Rôle of the pion electromagnetic form factor in the Δ(1232)→ γ ∗ N timelike transition
aa r X i v : . [ h e p - ph ] D ec Rˆole of the pion electromagnetic form factor in the ∆(1232) → γ ∗ N timelike transition G. Ramalho , M. T. Pe˜na , J. Weil , H. van Hees , and U. Mosel International Institute of Physics, Federal University of Rio Grande do Norte,Avenida Odilon Gomes de Lima 1722, Capim Macio, Natal-RN 59078-400, Brazil 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 Frankfurt Institute for Advanced Studies, Ruth-Moufang-Straße 1, D-60438 Frankfurt, Germany Institut f¨ur Theoretische Physik, Universit¨at Frankfurt,Max-von-Laue-Straße 1, D-60438 Frankfurt, Germany and Institut f¨ur Theoretische Physik, Universitaet Giessen, D35392- Giessen, Germany (Dated: October 16, 2018)The ∆(1232) → γ ∗ N magnetic dipole form factor ( G ∗ M ) is described here within a new covariantmodel that combines the valence quark core together with the pion cloud contributions. The pioncloud term is parameterized by two terms: one connected to the pion electromagnetic form factor,the other to the photon interaction with intermediate baryon states. The model can be used instudies of pp and heavy ion collisions. In the timelike region this new model improves the resultsobtained with a constant form factor model fixed at its value at zero momentum transfer. At thesame time, and in contrast to the Iachello model, this new model predicts a peak for the transitionform factor at the expected position, i.e. at the ρ mass pole. We calculate the decay of the ∆ → γN transition, the Dalitz decay (∆ → e + e − N ), and the ∆ mass distribution function. The impact ofthe model on dilepton spectra in pp collisions is also discussed. I. INTRODUCTION
To understand the structure of hadrons, baryons inparticular, in terms of quarks and gluons at low energies,is theoretically challenging due to the intricate combi-nation of confinement and spontaneous chiral symmetrybreaking, and the non-perturbative character of QCD inthat energy regime. Fortunately, experimentally electro-magnetic and hadron beams in accelerator facilities aredecisive tools to reveal that structure, and seem to in-dicate a picture where effective degrees of freedom asbaryon quark cores dressed by clouds of mesons play animportant role. For a review on these issues see [1]. Al-though different, experiments with electromagnetic andstrong probes complement each other. In electron scat-tering, virtual photons disclose the region of momentumtransfer q <
0, and spacelike form factors are obtained[1–3]. Scattering experiments of pions or nucleons withnucleon targets involving Dalitz decays of baryon reso-nances [2–5] provide information on timelike form factors,defined in the q > q = 0).Among the several baryon resonances the ∆ excita-tion and decays have a special role and are not yet fullyunderstood. The electromagnetic transition between thenucleon and the ∆(1232), and in particular its dominantmagnetic dipole form factor G ∗ M ( q ), as function of q , isa prime example that discloses the complexity of the elec- tromagnetic structure of the excited states of the nucleonand illustrates the limitations of taking into account onlyvalence quark degrees of freedom for the description ofthe transition.In the region of small momentum transfer G ∗ M ( q ) isusually underestimated by valence quark contributionsalone. Several models have been proposed in order tointerpret this finding. Most of them are based on the in-terplay between valence quark degrees of freedom and theso-called meson cloud effects, in particular, the dominantpion cloud contribution [1, 6–11]. Other recent works onthe ∆ → γ ∗ N transition can be found in Refs. [12–15].In this work we propose a hybrid model which com-bines the valence quark component, determined by a con-stituent quark model, constrained by lattice QCD andindirectly by experimental data, with a pion cloud com-ponent. The pion cloud component is written in terms ofthe pion electromagnetic form factor and therefore con-strained by data.The ∆ → γ ∗ N transition in the timelike region wasstudied using vector meson dominance (VMD) mod-els [16–19], the constant form factor model [5, 20], a twocomponent model (model with valence quark and me-son cloud decomposition), hereafter called the Iachellomodel [5, 21, 22], and the covariant spectator quarkmodel [4] (which incidentally also assumes VMD for thequark electromagnetic current).The Iachello model pioneered the timelike region stud-ies of the ∆ → γ ∗ N transition. The model was suc-cessful in the description of the nucleon form factors [21] ◆ ❇✶ ✁ ◆ ✁❇✷ ❇✸✭❛✮ ✭❜✮ FIG. 1: Pion cloud contributions for the ∆ → γ ∗ N electro-magnetic transition form factors. Between the initial and fi-nal state there are several possible intermediate octet baryonand/or decuplet baryon states: B in diagram (a); B and B in diagram (b). but has been criticized for generating the pole associ-ated with the ρ -meson pole near q ≃ . , below q = m ρ ≃ . [5] as it should. The constant formfactor model is a good starting point very close to q = 0but, on the other hand, does not satisfactorily take intoaccount the finite size of the baryons and their structureof non-pointlike particles.In the covariant spectator quark model the contribu-tions for the transition form factors can be separatedinto valence quark and meson cloud effects (dominatedby the pion). The valence quark component is directlyconstrained by lattice QCD data, and has been seen tocoincide with the valence quark core contributions ob-tained from an extensive data analysis of pion photo-production [8, 23, 24]. Its comparison to experimentaldata enables the extraction of information on the com-plementary meson cloud component in the spacelike re-gion [4, 6]. However the extension to the timelike re-gion of the meson cloud is problematic given the diffi-culty of a calculation that comprises also in a consistentway the whole meson spectrum. In Ref. [4] the mesoncloud was parameterized by a function F ρ , taken fromthe Iachello model where it describes the dressing of the ρ -propagator by intermediate ππ states. As noted be-fore, unfortunately, the function F ρ has a peak that isdisplaced relatively to the ρ -meson pole mass. Here, bydirectly using the pion form factor data we corrected forthis deficiency.Moreover, in previous works [4, 6–9] we have assumedthat the pion cloud contributions for the magnetic dipoleform factor could be represented by a simple parame-terization of one term only. But in the present workwe introduce an alternative parameterization of the pioncloud which contains two terms. These two leading ordercontributions for the pion cloud correspond to the two di-agrams of Fig. 1. We use then a parameterization of thepion cloud contributions for G ∗ M where diagram (a) isrelated to the pion electromagnetic form factor F π ( q ),and is separated from diagram (b). Diagram (a), wherethe photon couples directly to the pion, is dominant ac-cording to chiral perturbation theory, which is valid in the limit of massless and structureless quarks. But theother contribution, from diagram (b), where the photoncouples to intermediate (octet or decuplet) baryon stateswhile the pion is exchanged between those states, be-comes relevant in models with constituent quarks withdressed masses and non-zero anomalous magnetic mo-ments. This was shown in Ref. [10] on the study of themeson cloud contributions to the magnetic dipole mo-ments of the octet to decuplet transitions. The resultsobtained for the ∆ → γ ∗ N transition in particular, sug-gests that both diagrams contribute with almost an equalweight. II. IACHELLO MODEL
In the Iachello model the dominant contribution to the∆ → γ ∗ N magnetic dipole form factor is the meson cloudcomponent (99.7%) [5]. The meson cloud contributionsis estimated by VMD in terms of a function F ρ fromthe dressed ρ propagator, which in the limit q ≫ m π ,reads [4] F ρ ( q ) = m ρ m ρ − q − π Γ ρ m π q log q m π + i Γ ρ m π q , = m ρ m ρ + Q + π Γ ρ m π Q log Q m π . (2.1)In the previous equation Q = − q , m π is the pion mass,and Γ ρ is a parameter that can be fixed by the experimen-tal ρ decay width into 2 π , Γ ρ = 0 .
149 GeV or Γ ρ = 0 . III. COVARIANT SPECTATOR QUARKMODEL
Within the covariant spectator quark model frameworkthe nucleon and the ∆ are dominated by the S -wave com-ponents of the quark-diquark configuration [6, 25, 26].In this case the only non-vanishing form factor of the∆ → γ ∗ N transition is the magnetic dipole form factor,which anyway dominates in all circumstances.One can then write [6–8] G ∗ M ( q , W ) = G BM ( q , W ) + G πM ( q ) , (3.1)where G BM is the contribution from the bare core and G πM the contribution of the pion cloud. Here W generalizesthe ∆ mass M ∆ to an arbitrary invariant mass W in theintermediate states [4]. We omitted the argument W on G πM since we take that function to be independent of W .Following Refs. [4, 6–8] we can write G BM ( q , W ) = 83 √ MM + W f v ( q ) I ( q , W ) , (3.2)where I ( q , W ) = Z k ψ ∆ ( P + , k ) ψ N ( P − , k ) , (3.3)is the overlap integral of the nucleon and the ∆ radialwave functions which depend on the nucleon ( P − ), theDelta ( P + ) and the intermediate diquark ( k ) momenta.The integration symbol indicates the covariant integra-tion over the diquark on-shell momentum. For details seeRefs. [4, 6].As for f v ( q ) it is given by f v ( q ) = f − ( q ) + W + M M f − ( q ) (3.4)where f i − ( i = 1 ,
2) are the quark isovector form factorsthat parameterize the electromagnetic photon-quark cou-pling. The form of this parameterization assumes VMDmechanism [6, 25, 27]. See details in Appendix A.In this work we write the pion cloud contribution as G πM ( q ) = 3 λ π " F π ( q ) (cid:18) Λ π Λ π − q (cid:19) + ˜ G D ( q ) , (3.5)where λ π is a parameter that define the strength of thepion cloud contributions, F π ( q ) is a parameterizationof the pion electromagnetic form factor and Λ π is thecutoff of the pion cloud component from diagram (a).The function ˜ G D on Eq. (3.5) simulates the contribu-tions from the diagram (b), and therefore includes thecontributions from several intermediate electromagnetictransitions between octet and/or decuplet baryon states.From perturbative QCD arguments it is expected thatthe latter effects fall off with 1 /Q [28]. At high Q a baryon-meson system can be interpreted as a systemwith N = 5 constituents, which produces transitionform factors dominated by the contributions of the order1 / ( Q ) ( N − = 1 /Q . This falloff power law motivatesour choice for the form of ˜ G D : the timelike generaliza-tion of a dipole form factor G D = (cid:16) Λ D Λ D − q (cid:17) , where Λ D is a cutoff parameter defining the mass scale of the inter-mediate baryons.The equal relative weight of the two terms of Eq. (3.5),given by the factor λ π , was motivated by the resultsfrom Ref. [10], where it was shown that the contributionfrom each diagram (a) and (b) for the total pion cloudin the ∆ → γ ∗ N transition is about 50%. The overallfactor 3 was included for convenience, such that in thelimit q = 0 one has G πM (0) = 3 λ π . Since G ∗ M (0) ≃ λ π represents the fraction of the pion cloud contribution to G ∗ M (0).In the spacelike regime, in order to describe the va-lence quark behavior (1 /Q ) of the form factors associ-ated with the nucleon and ∆ baryons, the dipole formfactor G D with a cutoff squared value Λ D = 0 .
71 GeV had been used in previous works [6, 25]. As we will show,a model with Λ D = 0 .
71 GeV provides a very good de-scription of the ∆ → γ ∗ N form factor data in the region − < q <
0. However, since in the present workwe are focused on the timelike region, we investigate thepossibility of using a larger value for Λ D , such that theeffects of heavier resonances (Λ D ≈ ) can also betaken into account.To generalize G D to the timelike region we define˜ G D ( q ) ˜ G D ( q ) = Λ D (Λ D − q ) + Λ D Γ D , (3.6)where Γ D ( q ) is an effective width discussed in Ap-pendix B, introduced to avoid the pole q = Λ D . SinceΓ D (0) = 0, in the limit q = 0, we recover the spacelikelimit ˜ G D (0) = G D (0) = 1. We note that differently fromthe previous work [4] ˜ G D is the absolute value of G D ,and not its real and imaginary parts together.To summarize this Section: Eq. (3.5) modifies the ex-pression of the pion cloud contribution from our previousworks, by including an explicit term for diagram (b) ofFig. 1. Diagram (a) is calculated from the pion form fac-tor experimental data. Diagram (b) concerns less knownphenomenological input. The q dependence of that com-ponent is modeled by a dipole function squared. Since λ π was fixed already by the low q data, in the spacelikeregion, the pion cloud contribution is defined only by thetwo cutoff parameters Λ π and Λ D .Next we discuss the parameterization of the pion elec-tromagnetic form factor F π ( q ), which fixes the term fordiagram (a) and is known experimentally. IV. PARAMETERIZATION OF F π ( q ) The data associated with the pion electromagneticform factor F π ( q ) is taken from the e + e − → π + π − cross-section (the sign of F π ( q ) is not determined).The function F π ( q ) is well described by a simplemonopole form as F π ( q ) = αα − q − iβ , where α is a cut-off squared and β is proportional to a constant width.An alternative expression for F π ( q ), that replaces theIachello form F ρ is, F π ( q ) = αα − q − π βq log q m π + iβq . (4.1)Eq. (4.1) simulates the effect of the ρ pole with an effec-tive width regulated by the parameter β . Note that alsoEq. (4.1) has a form similar to the function F ρ of theIachello model given by Eq. (2.1). In particular, when α → m ρ and β → Γ ρ m π , we recover Eq. (2.1). The advan-tage of Eq. (4.1) over Eq. (2.1) is that α and β can beadjusted independently to the | F π | data. The result forthose parameters from the fit in both time- and spacelikeregions gives α = 0 .
696 GeV , β = 0 . . (4.2) -0.5 0 0.5 1 q (GeV ) | F π ( q ) | FIG. 2:
Fit to | F π ( q ) | data using Eq. (4.1). The data are fromRefs. [29, 30]. In the Iachello model (2.1) one has β ≃ .
1, a very differ-ent value. The fit is illustrated in Fig 2. The best fit se-lects α ≃ . , which is larger than m ρ ≃ . .However, in the best fit to the data, the value of α is corrected by the logarithmic counterterm in the denomi-nator of Eq. (4.1), that pushes the maximum of | F π ( q ) | to the correct position, q ≃ . . In the Iachellomodel, since β ≃ .
1, the correction is too strong, and themaximum moves to q ≃ . , differing significantlyfrom the | F π ( q ) | data.To describe the physics associated with the ρ -meson,we restricted the fit to q < . , which causes aless perfect description of F π at the right side of thepeak. However increasing q beyond that point slightlyworsens the fit. This probably indicates that althoughthe ω width is small, there may be some interferencefrom the ω mass pole, and that the parameters α and β account for these interference effects. Although thespacelike data was also included in the fit, the final re-sult is insensitive to the spacelike constraints. We obtainalso a good description of the spacelike region (exam-ine the region q < in Fig 2). The full exten-sion of the region where a good description is achieved is − < q < .A similar quality of the fit is obtained with both a con-stant width or a q -dependent ρ -width. However a betterfit can be obtained with a more complex q -dependence,which accounts better for the ω -meson pole effect, asshown in previous works [32, 33]. Since this work is meantto probe the quality of the results that one can obtainfor the transitions form factors, the simple analytic formof Eq. (4.1) suffices for F π ( q ).In addition, the covariant spectator quark model builtfrom this function describes well the ∆ → γ ∗ N form fac-tor in the spacelike region as shown in Fig. 3. Usingthe best fit of F π given by the parameters (4.2) we cancalculate the pion cloud contribution G πM ( q ) throughEq. (3.5), and consequently the result for G ∗ M ( q , M ∆ ).For the parameters λ π and Λ π we use the results ofthe previous works λ π = 0 .
441 and Λ π = 1 .
53 GeV , -1 -0.8 -0.6 -0.4 -0.2 0 q (GeV ) | G M * | Λ D2 = 0.71 GeV Λ D2 = 0.90 GeV FIG. 3: Results for | G ∗ M ( q ) | for the covariant spectatorquark model combined with the pion cloud contribution fromEq. (3.5). The data are from Refs. [31]. The dashed-dotted-line is the contribution from the core [4]. obtained from the comparison of the constituent quarkmodel to the lattice QCD data and experimental data [4,7, 8].In Fig. 3 we present the result of our model for | G ∗ M ( q , W ) | for the case W = M ∆ . In that case theimaginary contribution (when q >
0) is very small andthe results can be compared with the spacelike data( q < G BM ( q , M ∆ ) discussed in a previous work [4].In the same figure we show the sensitivity to the cutoffΛ D of the pion cloud model, by taking the cases Λ D =0 .
71 GeV and Λ D = 0 .
90 GeV . They are are consistentwith the data, although the model with Λ D = 0 .
71 GeV gives a slightly better description of the data. The twomodels are also numerically very similar to the results ofRef. [4] for W = M ∆ . For higher values of W the resultsof the present model and the ones from Ref. [4] will differ.Although the model with Λ D = 0 .
71 GeV gives a(slightly) better description of the spacelike data, for thegeneralization to the timelike region it is better to havea model with large effective cutoffs when compared withthe scale of the ρ meson pole (the ρ mass m ρ ). This isimportant to separate the effects of the physical scalesfrom the effective scales (adjusted cutoffs). V. RESULTS
The results for | G ∗ M ( q ) | from the covariant spectatorquark model for the cases W = 1 .
232 GeV, W = 1 . W = 1 . W = 2 . W are restrictedby the timelike kinematics through the condition q ≤ -1 -0.5 0 0.5 1 1.5 q (GeV ) | G M * | W = 1.232 GeVW = 1.600 GeVW = 1.800 GeVW = 2.200 GeV
FIG. 4: Results for | G ∗ M ( Q ) | for W = 1 .
232 GeV, W =1 . W = 1 . W = 2 . ( W − M ) , since the nucleon and the resonance (withmass W ) are treated both as being on their mass shells.Therefore the form factor covers an increasingly largerregion on the q axis, as W increases. See Ref. [4] for acomplete discussion.The figure illustrates well the interplay between thepion cloud and the bare quark core components. Thepion cloud component is dominating in the region nearthe ρ peak. Away from that peak it is the bare quarkcontribution that dominates. The flatness of the W =2 . q > is the net result of thefalloff of the pion cloud and the rise of the quark coreterms. In addition, the figure shows that dependence on W yields different magnitudes at the peak, and we recallthat this dependence originates from the bare quark corecontribution alone. This bare quark core contribution ismainly the consequence of the VMD parameterization ofthe quark current where there is an interplay between theeffect of the ρ pole and a term that behaves as a constantfor intermediate values of q (see Appendix A).We will discuss now the results for the widthsΓ γ ∗ N ( q, W ) of the ∆ Dalitz decay, and for the ∆ massdistribution g ∆ ( W ). A. ∆ Dalitz decay
The width associated with the ∆ decay into γ ∗ N canbe determined from the ∆ → γ ∗ N form factors for the∆ mass W . Assuming the dominance of the magneticdipole form factors over the other two transition formfactors, we can write [4, 5, 38]Γ γ ∗ N ( q, W ) = α
16 ( W + M ) M W × √ y + y − y − | G ∗ M ( q , W ) | , (5.1) q (GeV) -9 -8 -7 -6 -5 -4 -3 -2 -1 d Γ e + e - N / dq ( q , W ) W = 1.232 GeV W = 1.600 GeVW = 2.200 GeV
FIG. 5: Results for d Γ e + e − N dq ( q, W ) for three different valuesof energies W . The solid line is the result of our model. Thedotted line is the result of the constant form factor model. where q = p q , α ≃ /
137 is the fine-structure constantand y ± = ( W ± M ) − q .At the photon point ( q = 0), in particular, we obtainthe Γ γN in the limit q = 0 from Eq. (5.1) [5, 18, 37]Γ γN ( W ) = Γ γ ∗ N (0 , W ) . (5.2)We can also calculate the derivative of the Dalitz decaywidth Γ e + e − N ( q, W ) from the function Γ γ ∗ N ( q, W ) usingthe relation [5, 18, 37, 38]Γ ′ e + e − N ( q, W ) ≡ dΓ e + e − N d q ( q, W )= 2 α πq Γ γ ∗ N ( q, W ) . (5.3)The Dalitz decay width Γ e + e − N ( q, W ) is given byΓ e + e − N ( W ) = Z W − M m e Γ ′ e + e − N ( q, W ) d q, (5.4)where m e is the electron mass. Note that the integrationholds for the interval 4 m e ≤ q ≤ ( W − M ) , where thelower limit is the minimum value necessary to produce ane + e − pair, and ( W − M ) is the maximum value availablein the ∆ → γ ∗ N decay for a given W value.The results for dΓ e + e − N d q ( q, W ) for several mass values W (1.232, 1.6 and 2.2 GeV) are presented in Fig. 5. Theseresults are also compared to the calculation given by theconstant form factor model, from which they deviate con-siderably.Also, the ∆ decay width can be decomposed at treelevel into three independent channelsΓ tot ( W ) = Γ πN ( W ) + Γ γN ( W ) + Γ e + e − N ( W ) , (5.5)given by the decays ∆ → πN , ∆ → γN and ∆ → e + e − N . The two last terms are described respectively W (GeV) -8 -7 -6 -5 -4 -3 -2 -1 Γ ( G e V ) ∆ −> γ N ∆ −> e + e - N W (GeV) -8 -7 -6 -5 -4 -3 -2 -1 Γ ( G e V ) ∆ −> γ N ∆ −> e + e - N ∆ −> π N FIG. 6:
Results for the partial widths as function of W . At left: partial widths (solid line) for ∆ → γN and ∆ → e + e − N , comparedwith the constant form factor model (dotted line). At right: the partial widths are compared with the ∆ → πN width (dotted line) andwith the sum of all widths (thin solid line). W (GeV) -7 -6 -5 -4 -3 -2 -1 g ∆ ( W ) ( G e V - ) ∆ −> γ N ∆ −> e + e - N W (GeV) -7 -6 -5 -4 -3 -2 -1 g ∆ ( W ) ( G e V - ) ∆ −> π N ∆ −> γ N ∆ −> e + e - N FIG. 7: Results for g ∆ ( W ) and the partial contributions g ∆ → πN ( W ), g ∆ → γN ( W ) and g ∆ → e + e − N ( W ). At left: g ∆ → γN ( W )and g ∆ → e + e − N ( W ) in comparison with constant form factor model (dotted line). At right: all contributions compared withthe total g ∆ ( W ) (thin solid line). by Eqs. (5.2) and (5.4). The Γ πN term can be parame-terized as in [36, 39]Γ πN ( W ) = M ∆ W (cid:18) q π ( W ) q π ( M ∆ ) (cid:19) κ + q π ( M ∆ ) κ + q π ( W ) Γ πN , (5.6)where Γ πN is the ∆ → πN partial width for the physical∆, q π ( W ) is the pion momentum for a ∆ decay with mass W , and κ a cutoff parameter. Following Refs. [34, 35]we took κ = 0 .
197 GeV. The present parameterizationdiffers from other forms used in the literature [5, 37] andfrom our previous work [4].The results for the partial widths as functions of themass W are presented in Fig. 6. On the left panel wecompare Γ γN and Γ e + e − N with the result of the con-stant form factor model. On the right panel we presentthe total width Γ tot ( W ) as the sum of the three partialwidths. B. ∆ mass distribution To study the impact of the ∆ resonance propagationin nuclear reactions like the
N N reaction, it is necessaryto know the ∆ mass distribution function g ∆ ( W ). Asdiscussed before, W is an arbitrary resonance mass thatmay differ from the resonance pole mass ( M ∆ ). The usualansatz for g ∆ is the relativistic Breit-Wigner distribution-[4, 5] g ∆ ( W ) = A W Γ tot ( W )( W − M ) + W [Γ tot ( W )] , (5.7)where A is a normalization constant determined by R g ∆ ( W )d W = 1 and the total width Γ tot ( W ) (5.6).The results for g ∆ ( W ) and the partial contributions -3 -2 -1
0 0.2 0.4 0.6 0.8 1 d σ / d m ee [ µ b / G e V ] dilepton mass m ee [GeV]E kin = 1.25 GeV HADES dataGiBUU total ω → e + e - φ → e + e - ω → π e + e - π → e + e - γη → e + e - γ∆ QED ∆ RamalhoN* VMD ∆ * VMDBrems. OBE 0 0.2 0.4 0.6 0.8 1dilepton mass m ee [GeV]pp → e + e - XE kin = 2.2 GeV 0 0.2 0.4 0.6 0.8 1 10 -3 -2 -1 dilepton mass m ee [GeV]E kin = 3.5 GeV FIG. 8: Transport-model calculations of dilepton mass spectra d σ/ d m ee from proton-proton collisions ( pp → e + e − X ) at threedifferent beam energies, with and without a ∆ → γ ∗ N form factor, compared to experimental data measured with the HADESdetector [40–42]. g ∆ → γN ( W ) = Γ γN ( W )Γ tot ( W ) g ∆ ( W ) , (5.8) g ∆ → e + e − N ( W ) = Γ e + e − N ( W )Γ tot ( W ) g ∆ ( W ) , (5.9) g ∆ → πN ( W ) = Γ πN ( W )Γ tot ( W ) g ∆ ( W ) , (5.10)are presented in Fig. 7. The results are also comparedwith the constant form factor model. C. Dilepton production from NN collisions The ∆ → γ ∗ N magnetic dipole form factor in thetimelike region is known to have a significant influenceon dilepton spectra. Therefore we show in Fig. 8 atransport-model calculation of the inclusive dielectronproduction cross section d σ/ d m ee for proton-proton col-lisions (pp → e + e − X ), where m ee = q . These resultshave been obtained with the GiBUU model [34, 39] forthree different proton beam energies and are comparedto experimental data measured with the HADES detec-tor [40–42]. Except for the contribution of the ∆ Dalitzdecay, the calculations are identical to those presented inan earlier publication [35]. The ∆ Dalitz decay is shownin two variants, once with a constant form factor fixed atthe photon point (i.e., in ’QED’ approximation) and onceusing the form-factor model described in the precedingsections.At the lowest beam energy of 1.25 GeV, the produced∆ baryons are close to the pole mass and therefore theresults with and without the form factor are very sim-ilar. At higher beam energies, however, the model forthe ∆ → γ ∗ N form factor has a much larger impact, be-cause higher values of W are reached, where the formfactor deviates strongly from the photon point value.In Fig. 9 we illustrate the influence of W by showing the W distribution of produced ∆ + , baryons in theGiBUU simulations. We note that several different pro-cesses contribute to the inclusive ∆ + , production, suchas N N → N ∆ , ∆∆ , ∆ N ∗ etc., each of which will pro-duce a different W distribution due to different kinemat-ics and phase space. Furthermore it should be remarkedthat the tails of this distribution, just as the ∆ spectralfunction in Eq. (5.7), depend significantly on the specificparameterization of the hadronic width for ∆ → πN .However, for electromagnetic observables as shown inFig. 8, the dependence on the hadronic width is veryweak, since in Eq. (5.9) the total width cancels out inthe numerator and only stays in the denominator. d σ / d W [ m b / G e V ] W [GeV]pp → ∆ +,0 XE kin = 1.25 GeVE kin = 2.2 GeVE kin = 3.5 GeV FIG. 9: Mass distribution of produced ∆ + , baryons inGiBUU simulations, for pp collisions at three different beamenergies. -3 -2 -1
0 0.2 0.4 0.6 0.8 1 d σ / d m ee [ µ b / G e V ] dilepton mass m ee [GeV]E kin = 1.25 GeV HADES dataGiBUU total ω → e + e - φ → e + e - ω → π e + e - π → e + e - γη → e + e - γ∆ QED ∆ RamalhoN* VMD ∆ * VMDBrems. OBE 0 0.2 0.4 0.6 0.8 1dilepton mass m ee [GeV]pp → e + e - XE kin = 2.2 GeV 0 0.2 0.4 0.6 0.8 1 10 -3 -2 -1 dilepton mass m ee [GeV]E kin = 3.5 GeV FIG. 10: Modified calculations of dilepton mass spectra d σ/ d m ee from proton-proton collisions (pp → e + e − X ), using reduced R → ρN branching ratios for two resonances (see text). Coming back to Fig. 8, it should be noted that thechoice of the form factor has little influence on the overallagreement of the total dilepton spectrum with the experi-mental data at the two lowest beam energies, because theinfluence of the form factor is weak or the ∆ contributionis small compared to other channels. At the highest beamenergy of 3.5 GeV, however, the choice of the form factordoes have an impact on the total spectrum for massesabove 600 MeV. While the constant-form-factor resultcombined with the other channels from GiBUU shows agood agreement with the data, using the q dependentform factor results in a slight overestimation of the data,which is most severe for masses of around 700 MeV. How-ever, we note that the ∆ contribution by itself does notovershoot the data. Only in combination with the otherchannels (in particular the heavier baryons, such as N ∗ and ∆ ∗ ) the overestimation is seen.There could be various reasons for this enhancementover the data, but we want to mention here only the twomost likely ones. One could lie in the form factor itself,more precisely in the omission of an W dependence ofthe overall weight λ π for the pion cloud. This parameterfor the weight of the pion cloud should probably dependon W . If the two diagrams (a) and (b) of the pion cloudcontribution would decrease simultaneously with W , aswe can expect from the drop of the m π /W ratio, thiscould potentially cure the observed overestimation.On the other hand, the reason for the disagreementcould also be found in the other channels that are part ofthe transport calculation. In particular the contributionsof the higher baryonic resonances ( N ∗ and ∆ ∗ ) are sub-ject to some uncertainties. These resonance contributionswere recently investigated via exclusive pion productionat 3.5 GeV with the HADES detector [43], which showedthat the GiBUU model does a rather good job in de-scribing the resonance cocktail for the exclusive channels(with some minor deviations). However, there are alsosignificant non-exclusive channels for pion and dilepton production at this energy. Moreover, the form factors ofthe higher resonances are a matter of debate (they aretreated in a strict-VMD assumption in the calculation).It was remarked in [43] that some of the branchingratios for R → ρN , which directly influence the dilep-ton yield via the VMD assumption, might be overesti-mated in GiBUU, in particular for the N ∗ (1720) and the∆ ∗ (1905). Both have a very large ρN branching ratio of87% in GiBUU [34] (as adopted from [36]) and also in thecurrent PDG database these branching ratios are listedwith rather large values [44], which are essentially com-patible with the GiBUU values. However, some recentpartial-wave analyses [45, 46] claim much smaller valuesfor these branching ratios, showing some tension withthe PDG and GiBUU values. We show in Fig. 10 the ef-fect of using smaller values for these branching ratios onthe dilepton spectra, adopting the upper limits from theBonn-Gatchina analysis [45] (as given in [43]), namely10% for N ∗ (1720) → ρN and 42% for ∆ ∗ (1905) → ρN .We note that the values in [46] are even smaller. As seenin Fig. 10, this change indeed reduces the contributionsfrom the N ∗ and ∆ ∗ resonances by a fair amount, in par-ticular in the high-mass region ( m ee >
600 MeV). Thisimproves the agreement with the highest data points at2.2 GeV, and it also mitigates the overshooting over thedata at 3.5 GeV when the ∆ → γ ∗ N form factor is used,but it does not fully cure it.Thus it is quite likely that the remaining excess iscaused by the negligence of the W dependence in the pioncloud contribution of the form factor. A more detailedinvestigation of the W dependence of the pion cloud isplanned in a further study that will analyze all these as-pects. VI. SUMMARY AND CONCLUSIONS
In this work we present a new covariant model for the∆ → γ ∗ N transition in the timelike region. The modelis based on the combination of valence quark and mesoncloud degrees of freedom. The bare quark contributionwas calibrated previously to lattice QCD data. One ofthe pion cloud components is fitted to the pion electro-magnetic form factor F π (with the fit being almost in-sensitive to the spacelike data and strongly dependenton the timelike data) and the other, associated with in-termediate octet/decuplet baryon states, parameterizedby an effective cutoff Λ D .Our model induces a strong effect on the ∆ → γ ∗ N magnetic dipole form factor in the region around the ρ − meson pole (where the magnitude is about four timeslarger than at q = 0). This effect was missing in thefrequently used Iachello model. The pion cloud effectsdominate in the region q ≤ . . For larger q the effects of the valence quark became dominant, andthe q -dependence is smoother. At low energies, the newform factor has little influence on the overall agreementof the total dilepton spectrum in N N collisions with theexperimental data, and no large difference between ournew model and the VMD model is seen. However at thehighest beam energy of 3.5 GeV, the choice of the formfactor does affect the total spectrum for masses above600 MeV.Measurements of independent channels, for instanceexclusive pion induced ∆ production data, can help tobetter constrain the pion cloud contribution. The meth-ods presented in this work can in principle be extendedto higher mass resonances as N ∗ (1440), N ∗ (1520), N ∗ (1535), N ∗ (1710) and ∆ ∗ (1600), for which there arealready predictions of the covariant spectator quarkmodel [47, 48] in the spacelike region. The calculationof the N ∗ (1520) form factors in the timelike region [49],extending the results from Ref. [47] is already under way. Acknowledgments
The authors thank Marcin Stolarski and Elmar Bier-nat for the information about the pion electromag-netic form factors. G.R. was supported by the Brazil-ian Ministry of Science, Technology and Innovation(MCTI-Brazil). M.T.P. received financial support fromFunda¸c˜ao para a Ciˆencia e a Tecnologia (FCT) un-der Grants Nos. PTDC/FIS/113940/2009, CFTP-FCT(PEst-OE/FIS/U/0777/2013) and POCTI/ISFL/2/275.This work was also partially supported by the Euro-pean Union under the HadronPhysics3 Grant No. 283286.J.W. acknowledges funding of a Helmholtz Young Investi-gator Group VH-NG-822 from the Helmholtz Associationand GSI.
Appendix A: Quark form factors
We use a parameterization of the quark isovector formfactors motivated by VMD [8, 23, 25] f − ( q ) = λ q + (1 − λ q ) m ρ m ρ − q − c − M h q ( M h − q ) f − ( q ) = κ − ( d − m ρ m ρ − q + (1 − d − ) M h M h − q ) , (A1)where m ρ = 775 MeV is the ρ -meson mass, M h is themass of an effective heavy vector meson, κ − is the quarkisovector anomalous magnetic moment, c − , d − are mix-ture coefficients, and λ q is a parameter related with thequark density number in the deep inelastic limit [25]. Theterm in M h , where M h = 2 M , simulates the effects of theheavier mesons (short range physics) [25], and behaves asa constant for values of q much smaller than 4 M . Thewidth associated with the pole q = M h is discussed inthe Appendix B.The ρ pole appears when one assumes a stable ρ withzero decay width Γ ρ = 0. For the extension of the quarkform factors to the timelike regime we consider thereforethe replacement m ρ m ρ − q → m ρ m ρ − q − i m ρ Γ ρ ( q ) . (A2)On the r.h.s. we introduce Γ ρ the ρ decay width as afunction of q .The function Γ ρ ( q ) represents the ρ → π decay widthfor a virtual ρ with momentum q [32, 50]Γ ρ ( q ) = Γ ρ m ρ q (cid:18) q − m π m ρ − m π (cid:19) θ ( q − m π ) , (A3)where Γ ρ = 0 .
149 GeV.
Appendix B: Regularization of high momentumpoles
For a given W the squared momentum q is limited bythe kinematic condition q ≤ ( W − M ) . Then, if one hasa singularity at q = Λ , that singularity will appear forvalues of W such that Λ ≤ ( W − M ) , or W ≥ M + Λ.To avoid a singularity at q = Λ , where Λ is anyof the cutoffs introduced in our pion cloud parameteri-zations, and quark current (pole M h ) we implemented asimple procedure. We start withΛ Λ − q → Λ Λ − q − iΛΓ X ( q ) , (B1)where Γ X ( q ) = 4Γ X (cid:18) q q + Λ (cid:19) θ ( q ) , (B2)0In the last equation Γ X is a constant given by Γ X =4Γ ρ ≃ . X ( q ) is defined such thatΓ X ( q ) = 0 when q <
0. Therefore the results in thespacelike region are kept unchanged. For q = Λ weobtain Γ X = Γ X , and for very large q it follows Γ X ≃ X . Finally the value of Γ X was chosen to avoid verynarrow peaks around Λ .While the width Γ ρ ( q ) associated with the ρ -mesonpole in the quark current is nonzero only when q > m π ,one has for Γ X ( q ) nonzero values also in the interval4 m π > q >
0. However, the function Γ X ( q ) changes smoothly in that interval and its values are negligible.This procedure was used in Ref. [4, 34] for the cal-culation of the ∆ → γ ∗ N form factors in the timelikeregime. In the present case the emerging singularitiesfor W > M + Λ D ≃ .
84 GeV are avoided, and for
W < .
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