Reggeized model for γp \to ρ^- Δ^{++}(1232) photoproduction
aa r X i v : . [ h e p - ph ] D ec The Reggeized model for γp → ρ − ∆ ++ (1232) photoproduction Byung-Geel Yu ∗ and Kook-Jin Kong † Research Institute of Basic Sciences, Korea Aerospace University, Goyang, 412-791, Korea (Dated: March 13, 2018)We construct a model for the reaction process γp → ρ − ∆ ++ by utilizing the Reggeization of the t -channel meson exchange and present the analysis of the existing data at high energies. Based onthe simple π + ρ exchanges where the t -channel ρ exchange is conserved with the u -channel ∆-polein addition to the s -channel proton pole and the contact term, we discuss the convergence of thereaction cross section at high energy in association with the gauge prescription for the u -channel∆-pole as well as the proton-pole in the s -channel. The roles of the electromagnetic (EM) multipolemoments of the ∆-baryon and ρ -meson are analyzed. The agreement of the present calculation withthe data of total and differential cross sections and spin density matrix elements for the γp → ρ − ∆ ++ process are shown. Model predictions for the measurements of the electromagnetic moments of the ρ and ∆ are given to the photon polarization asymmetry. PACS numbers: 11.55.Jy, 13.60.Rj, 13.60.Le, 13.85.Fb, 14.20.Gk, 14.40.BeKeywords: ∆(1232) and ρ photoproduction, Gauge invariance Electromagnetic multipole moments, Reggetrajectory Photoproduction of charged ρ with ∆-baryon in thefinal state is one of the issues which have been less chal-lenged despite its significance to our understanding of theinteraction between hadrons of higher spins.To date, however, few theoretical investigations arefound in literature to treat hadron productions of suchhigh-spins by electromagnetic probe, though the en-ergy, angle and polarization dependences of the reac-tion cross sections had been measured for the reaction γp → ρ − ∆ ++ in various photon energies more than fortyyears ago [1][2][3–6].With empirical evidences of ρ ∆ formation in the fi-nal state from the analysis of the reaction process γp → pπ + π π − , a theoretical study of the γp → ρ − ∆ ++ pro-cess was attempted in Ref. [7], but limited only to aconstruction of the Born approximation amplitude witha special gauge prescription applied for the ρ exchange.The analysis of the γp → ρ − ∆ ++ process by using the π + b + ρ + a Regge-poles to fit to data in the s -channelhelicity amplitude was presented in Ref. [8]. A quali-tative analysis of the process γp → ρ ± ∆ is made for asubprocess in the π photoproduction in Ref. [9]. How-ever, all these are not complete to fully elucidate theproduction mechanism of such high-spin ∆-baryon withthe ρ from the standpoint of the effective Lagrangian for-mulation.In previous works we investigated photoproductionof spin-1 vector meson on nucleon, γN → ρ ± N and γN → K ∗ Λ [10, 11], and that of spin-3/2 ∆-baryon γp → π ± ∆ [12] by using the Born amplitude where the t -channel meson exchanges were reggeized following theprocedures of Ref. [13]. In all these reactions the energy-dependence of cross sections showed a common featureof the nondiffractive two-body scattering with a sharp ∗ E-mail: [email protected] † E-mail: [email protected] peaking behavior near threshold. From the standpointof the Regge formalism, this can be understood by therelation, σ ∼ s α J (0) − , which predicts the dominance ofthe π exchange over the ρ (and K over K ∗ ) to agreewith the steep decrease observed in the total cross sec-tion beyond the resonance region. Indeed, our previousanalyses on these photoproduction processes showed anagreement with such a production mechanism withouteither fit-parameters or ad hoc counter terms considered.On the other hand, the study of the former two processesserved us to provide information on the EM multipolemoments of the charged vector meson ρ ( K ∗ ), and thecase of the π ∆ process led us to consider a special gaugeprescription (the minimal gauge) in order for a conver-gence of the cross section at high energy [14].Hinted by these findings, we here investigate the γp → ρ − ∆ ++ process as a natural extension of our previousworks, and our theoretical interest in the present issue istwo-folded; the role of the EM multipole moments of ρ -meson and ∆-baryon, and the convergence of the reactioncross section at high energy because both the propagatorsof the spin-1 and spin-3/2 particles have the term pro-portional to p /M which would diverge at high energy,unless one makes an approximation. The process with full propagations of ∆ and ρ :Model I Since a particle of spin- J has 2 J + 1 EM multipole mo-ments, there are four EM multipole moments at the γ ∆∆vertex in addition to the three multipole moments at the γρρ vertex in the charged process γp → ρ − ∆ ++ . Thiscould cause further complication in establishing gaugeinvariance of the reaction process. As discussed in Ref.[10], the validity of the Ward identity for the γρρ cou-pling is crucial to render gauge invariance rather a sim-pler form. This is true for the γ ∆∆ vertex as well [12].In this work, therefore, based on the Ward identities forTypeset by REVTEX N ∆ λ ρ ( η α ) γ ( ǫ µ ) ρ ( η ν ) N N ∆ λ γ ( ǫ µ ) ρ ( η ν ) N ∆ β ∆ λ γ ( ǫ µ ) ρ ( η ν ) γ ( ǫ µ ) ρ ( η ν ) N ∆ λ ( a ) ( b )( c ) ( d ) ρ ( η β )∆ σ FIG. 1. Feynman diagrams for the gauge-invariant ρ exchangein γ ( k ) + N ( p ) → ρ ( q ) + ∆( p ′ ). π and other meson exchangesproceed via the t -channel exchange (a). the charge couplings in these vertices we first formulatea gauge invariant model for the γp → ρ − ∆ ++ processwith the full propagators taken into account for boththe ∆-baryon and ρ -meson. This will make the model-description complete for the interaction between higher-spin particles with the EM multipole moments fully con-sidered. We, then, proceed to the higher energy region toinvestigate the convergence of the reaction cross section.For the photoproduction process γ ( k ) + p ( p ) → ρ − ( q ) + ∆ ++ ( p ′ ) , (1)where the momenta of the initial photon, nucleon and thefinal ρ and ∆ are denoted by k , p , q , and p ′ , respectively,the current conservation following the charge conserva-tion, e p − e ρ − − e ∆ ++ = 0, requires the ∆-pole in the u -channel in addition to the s -channel proton-pole andthe contact term. Thus, we have the gauge-invariant ρ exchange in the t -channel which is given by M ρ = u λ ( p ′ ) η ∗ ν ( q ) × h M λνµt ( ρ ) + M λνµs ( p ) + M λνµu (∆) + M c i ǫ µ ( k ) u ( p ) . (2)Here u λ ( p ′ ), u ( p ), η ν ( q ), and ǫ µ ( k ) are the spin-3/2 ∆-spinor of the Rarita-Schwinger field, Dirac spinor for nu-cleon, and the spin polarizations of ρ and photon. InEq. (2) the respective particle exchanges and the con-tact term are given by M λνµs ( p ) = Γ λνρN ∆ ( q, p ′ , p + k ) / p + / k + M N s − M N Γ µγNN ( k ) , (3) M λνµt ( ρ ) = Γ νµαγρρ ( q, Q ) − g αβ + Q α Q β /m ρ t − m ρ Γ βλρN ∆ ( Q, p ′ , p ) , (4) M λνµu (∆) = Γ λµσγ ∆∆ ( k ) / p ′ − / k + M ∆ u − M Π ∆ σβ ( p ′ − k ) × Γ βνρN ∆ ( q, p ′ − k, p ) , (5) and M c = − u λ ( p ′ ) (cid:26) e ρ f ρN ∆ m ρ ( ǫ λ / η ∗ − η ∗ λ / ǫ )+ g ρN ∆ m ρ [ e ∆ ( q λ ǫ · η ∗ − η ∗ λ ǫ · q ) + e ρ ( ǫ λ p ′ · η ∗ − η ∗ λ ǫ · p ′ )]+ h ρN ∆ m ρ [ e N ( q λ ǫ · η ∗ − η ∗ λ ǫ · q ) + e ρ ( ǫ λ p · η ∗ − η ∗ λ ǫ · p )] (cid:27) × γ u ( p ) (6)with Q µ = ( q − k ) µ , the t -channel momentum transferand the spin-3/2 projection which is given byΠ ∆ µν ( p ) = − g µν + γ µ γ ν γ µ p ν − γ ν p µ M ∆ + 2 p µ p ν M . (7)The electromagnetic coupling vertices γN N , γ ∆∆ [15]and γρρ [10] which fully account for their EM multipolemoments are defined as follows, ǫ µ Γ µγNN ( k ) = e N / ǫ − eκ N M N [/ ǫ, / k ] , (8) ǫ µ Γ λµσγ ∆∆ ( p ′ , k, p ) = − (cid:26) e ∆ ( g λσ / ǫ − ǫ λ γ σ − γ λ ǫ σ + γ λ / ǫγ σ ) − e M ∆ (cid:18) κ ∆ g λσ + χ ∆ k λ k σ M (cid:19) [/ ǫ, / k ]+ eλ ∆ M (cid:20) k λ k σ / ǫ −
12 / k ( ǫ λ k σ + ǫ σ k λ ) (cid:21)(cid:27) , (9)Γ νµαγρρ ( q, Q ) ǫ µ = − e ρ (cid:26)(cid:20) ( q + Q ) µ g να − Q ν g µα − q α g µν (cid:21) + κ ρ ( g µα k ν − g µν k α ) − ( λ ρ + κ ρ )2 m ρ (cid:20) ( q + Q ) µ k ν k α −
12 ( q + Q ) · k ( k ν g µα + k α g µν ) (cid:21)(cid:27) ǫ µ . (10)Here e ∆ , κ ∆ , χ ∆ , and λ ∆ are the charge, anoma-lous magnetic moment, magnetic octupole, and electricquadrupole moments of the ∆, respectively. e ρ , κ ρ and λ ρ are the charge, anomalous magnetic moment and elec-tric quadrupole moment of ρ -meson.Note that, in particular, the charge-coupling terms inEqs. (8), (9), and (10) satisfy the Ward identities in theirrespective vertices [10, 12].For the strong coupling vertex ρN ∆ we utilized thefollowing form from Ref. [7]Γ λνρN ∆ ( q, p ′ , p ) = (cid:20) f ρN ∆ m ρ (cid:0) q λ γ ν − / qg λν (cid:1) + g ρN ∆ m ρ (cid:0) q λ p ′ ν − q · p ′ g λν (cid:1) + h ρN ∆ m ρ (cid:0) q λ p ν − q · p g λν (cid:1) (cid:21) γ (11)with the quark model prediction, f ρ − p ∆ ++ = √ f ρ pp =8 .
57, which is from the relation f ρ pp m ρ = g ρ pp M (1 + κ ρ )for the ρ exchange in the NN potential [16] and using g ρ pp = 2 . κ ρ = 3 . f ρN ∆ was determined tobe smaller than the one discussed above we take the f ρN ∆ constant in the range from 4.7 to 8.6 in the calculation.For a consistency with the previous work on γN → ρ ± N the EM multipole moments of ρ -meson, κ ρ and λ ρ ,are taken the same as in Ref. [10]. The values for the EMmultipole moments of the ∆-baryon in Eq. (9) are takenfrom Ref. [15] for a complete set of these observables.In the formal reggeization the gauge-invariant ρ ex-change is now written as M ρ = M ρ × ( t − m ρ ) R ρ ( s, t ) , (12)where R ϕ ( s, t ) = πα ′ ϕ × phaseΓ( α ϕ ( t ) + 1 − J ) sin πα ϕ ( t ) (cid:18) ss (cid:19) α ϕ ( t ) − J (13)is the Regge-pole written collectively for a meson ϕ ofspin- J with the phase and trajectory α ( t ) J [10]. Themass parameter is taken to be s =1 GeV .The gauge-invariant pion exchange in the t -channel isgiven by, M π = − i g γπρ m f πN ∆ m π ε µναβ ǫ µ η ∗ ν k α Q β u λ ( p ′ ) Q λ u ( p ) ×R π ( s, t ) , (14)where g γπρ = ± .
224 is estimated from the measureddecay width and f π − p ∆ ++ is chosen in the range from 1 . .
16 [12] for a better agreement with existing data.By using the coupling vertex a N ∆ with the couplingconstant f a N ∆ m a = − f ρN ∆ m ρ as in Ref. [12], and the vertex γρa in Ref. [19] with the coupling constant 0 .
044 GeV − determined from the decay width [20] we estimate thecontribution of the a exchange to find it to be of the10 − order. Hence, it is neglected in this work.Table I summarizes the physical constants for theModel I and Model II which are referred in advance forlater use.We now discuss the determination of the phases of π and ρ exchanges. Given the isospin relation for the dif-ferent charge states of ρN ∆ couplings f ρ − p ∆ ++ = f ρ + n ∆ − = √ f ρ + p ∆ = √ f ρ − n ∆ + , (15)and from the G -parity for photon-meson coupling whichdictates the signs of ρ and b [10] to be changed in accordwith their charges, we write the production amplitudesrelevant to the present issue, i.e., M γp → ρ − ∆ ++ = − M ρ + M a + M π − M b , (16) M γn → ρ + ∆ − = M ρ + M a + M π + M b . (17)In the absence of the a and b further, the situation issimilar to the case of γN → ρ ± N [10]. In order to obtaina fair description of existing data up to E γ ≈
10 GeV,where the dominance of π exchange over the ρ is required TABLE I. Physical constants in this work. Proton anoma-lous magnetic moment κ p = 1 . ρ -meson EM multipolemoments κ ρ = 1 .
01 and λ ρ = − .
41 are fixed in both models.The effects from the multipole moments denoted as starred, λ ∆ ++ and χ ∆ ++ are considered only in the limited energy re-gion as in Fig. 2 (b). EM multipoles of ∆-baryon are takenfrom Ref. [15]. Model I Model II κ ∆ ++ .
34 0 λ ∆ ++ . ∗ χ ∆ ++ . ∗ f ρN ∆ g ρN ∆ h ρN ∆ − g γπρ ± .
224 0.224 f πN ∆ for the steep decrease of the cross section we employ thephases given as follows, M ( ρ ∓ ) = ∓ ρ ×
12 ( − e − iπα ρ ( t ) ) + π × ( e − iπα π ( t ) ) . (18)In Fig. 2 we present the result of the Model I in the -1 (f ρ N ∆ = 5.5, g ρ N ∆ = 0, h ρ N ∆ = 0) E γ [GeV] E γ [GeV] σ [µ b] ( γ p → ρ − ∆ ++ ) κ ∆ =0, λ ∆ =0, χ ∆ =0κ ∆ =4.34, λ ∆ =6.18, χ ∆ =12.34 (f ρ N ∆ = 5.5, g ρ N ∆ = 0, h ρ N ∆ = −2) κ ∆ =4.34, λ ∆ =0, χ ∆ =0 Π µν∆ = - g µν M ρ (a) (b) FIG. 2. Total cross section for γp → ρ − ∆ ++ from the ModelI. (a): The cross section of the solid line shown in the log-scalebecomes divergent above E γ ≃
10 GeV due to the divergenceof the u -channel ∆ over E γ ≃ ρ exchange M ρ in Eq. (12) which is depicted by the reddash-dotted line. The nonvanishing h ρN ∆ -coupling plays therole to suppress the divergence of the cross section up to ≈
10 GeV. The approximation of Π ∆ µν ≈ − g µν shows a largedeviation with data. (b): Roles of the ∆-baryon EM momentsare shown in the total cross section. The cross section withEM moments fully accounted diverges over 4 GeV but yieldsthe finite result below the energy region. Data are taken fromRefs. [1, 3–5]. total cross section from the ρ + π exchanges with thetrajectories α ρ ( t ) = 0 . t − m ρ ) + 1 , (19) α π ( t ) = 0 . t − m π ) , (20)taken and physical constants from Table I. In both panels(a) and (b) the solid line represents the total cross sectionfrom the calculation with κ p , κ ∆ ++ , κ ρ and λ ρ turned onin addition to their charges. Based on this, the cases ofthe vanishing constants g ρN ∆ = h ρN ∆ = 0, and the ap-proximation of Π ∆ µν ≈ − g µν for the ∆-propagation aretested, and they are found to be valid only in the lim-ited range of energy as denoted by the red dashed, andblue dotted lines, respectively. In (b) the sensitivity ofthe cross section to the EM multipole moments of ∆ isexamined and the role of κ ∆ is found to be of significanceas can be expected from the magnetic nature of the ∆.Nevertheless, however, the prediction of the Model I withthe EM multipole moments of ∆ fully accounted wouldfall down above E γ ≃ ρ is muchless significant than that of ∆.In the scheme where the spin-3/2 polarization tensorΠ ∆ µν is fully considered for the ∆ propagation in the u -channel, we note that the cross section of the Model Ias shown by the solid line is valid up to E γ ≃
10 GeVwith the energy-dependence finite. But it becomes diver-gent due to the u -channel divergence from the ∆, thoughgauge-invariant.In comparison to the γp → π − ∆ ++ process where thecross section already diverges about E γ ≃ . ρN ∆ with 5 . /m ρ than the πN ∆ coupling with 1 . /m π to the ∆-pole in the u -channel.In the next, let us consider the minimal gauge prescrip-tion for the conserved ρ exchange in order to render thecross section convergent far beyond the energy. The minimal gauge prescription: Model II
We apply the minimal gauge prescription for gauge in-variance of the ρ exchange, where the proton and ∆ polesin the s - and u -channels are introduced in the minimalway, i.e., only the non-gauge-invariant remnant of the s -and u -channel electric Born terms are indispensable tocompensate for the lack of the t -channel ρ exchange inthe current conservation [14].The gauge-invariant terms to be removed by redun-dancy is easily checked by replacing ǫ with k and usingthe on-shell conditions u λ ( p ′ ) γ λ = 0 in the EM verticesEqs. (8) and (9). Then, similar to the π ∆ case as dis-cussed in Ref. [12], the u -channel ∆-pole as in Eq. (23)given below is obtained from such an antisymmetric formof the u -channel amplitude as in Eq. (12) of Ref. [12]due to the term − ǫ λ γ σ in the γ ∆∆ vertex. -1 γ [GeV] E γ [GeV] σ [µ b] ( γ p → ρ − ∆ ++ ) Model II σ [µ b] ( γ p → ρ − ∆ ++ )πρ (a) (b)M ρ FIG. 3. Total cross section for γp → ρ − ∆ ++ from the ModelI and II. (a): Contributions of the π and ρ exchanges in the t -channel and that of the M ρ are given for comparison. (b):The cross section from the Model II shows a good convergenceat high energy. The dotted line is the cross section without ρ -meson EM multipole moments κ ρ and λ ρ . Notations forcurves are the same as in the panel (a). Data are taken fromRefs. [1, 3–5]. The minimal gauge-invariant ρ exchange in Eq. (2) isnow expressed as, M λνµs ( p ) = f ρN ∆ m ρ (cid:0) q λ γ ν − / qg λν (cid:1) γ p µ s − M N e N , (21) M λνµt ( ρ ) = Γ νµαγρρ ( q, k, Q ) − g αβ + Q α Q β /m ρ t − m ρ × f ρN ∆ m ρ (cid:0) Q λ γ ν − / Qg λν (cid:1) γ , (22) M λνµu (∆) = e ∆ p ′ µ u − M f ρN ∆ m ρ (cid:0) q λ γ ν − / qg λν (cid:1) γ , (23)with the contact term given in Eq. (6). A few remarksare in order: In the minimal gauge we neglect the termsof g ρN ∆ and h ρN ∆ couplings because of no need to sup-press the divergence of the ∆-pole. Strictly speaking,the γρρ vertex in Eq. (22) should also be simplified tohave only the charge-coupling terms. Nevertheless we re-sume the original form to investigate the effects of theEM multipole moments.Figure 3 shows the comparison of the cross sectionsbetween Model I and II with the contributions of me-son exchanges therein. In Fig. 3 (b) we demonstrate agood convergence of the cross section at high energy bythe dominating role of the π exchange over the ρ in theminimal gauge. In order to yield a better result from theModel II we use the coupling constants f πN ∆ = 2 and f ρN ∆ = 8 .
57 which were adopted in the γp → π − ∆ ++ process [12]. We show the effect of the ρ -meson EM mul-tipole moments κ ρ and λ ρ by the solid and dashed linesin (b).Differential cross sections and density matrix elements -1 d σ / d t [ µ b / G e V ] -1 ]10 -1 d σ / d t [ µ b / G e V ] ]10 -3 -2 -1 E γ = 3.15 GeV E γ = 3.85 GeVE γ = 4.5 GeV E γ = 9.6 GeV γ p -> ρ − ∆ ++ I πρ II FIG. 4. Differential cross sections for γp → ρ − ∆ ++ . Contri-butions of π and ρ exchanges are shown in the dashed anddash-dotted lines at E γ = 3 .
15 GeV. The cross sections fromthe Model I (dotted) and II (solid line) are presented for com-parison. Data are taken from Refs. [1, 2]. for the unpolarized process are given in Figs. 4 and 5,respectively. The contributions of π and ρ exchanges areshown at E γ = 3 .
15 GeV. The difference of the modelpredictions between I and II is presented for comparison.The angular dependence of the differential cross section isin fair agreement with the canonical phase taken for the ρ exchange which reproduces the dip at − t ≈ . by the nonsense-wrong-signature-zero at α ρ ( t ) = 0.The density matrix elements presented in Fig. 5 arecalculated in the Gottfried-Jackson (G.-J.) frame wherethe ρ → ππ decay with the ∆-baryon in the final state isdeveloped by following the conventions and definitions ofRef. [21]. Due to the canonical phase of the ρ exchangesuch an oscillatory behavior as in Fig. 5 could describethe t -dependence of the density matrix elements to somedegree.Finally, with an expectation of the magnetic natureof both the ∆ and ρ in the interaction with photon wemake a prediction for the measurement of the magneticmoments of ∆ and ρ from the photon polarization asym-metry for future experiments. Figure 6 presents the pho-ton polarization asymmetry Σ with and without κ ∆ withrespect to the angle (a) and energy (b), respectively.Likewise, the cases of the ρ with and without the EMmultipole moments are given in the raw (c) and (d).To summarize, we have investigated the photoproduc-tion γp → ρ − ∆ ++ process with a particular interest inthe role of the ρ and ∆-baryon EM multipole momentsas well as the convergence of the cross section at highenergy. By constructing two versions in order to treatthe divergence of the ∆-propagation in the u -channel,the so-called Model I and II, we examined the energy de-pendence of the cross section up to E γ = 10 GeV and 16GeV, respectively. The important findings in the presentwork are as follows: We observed a strong peak in the -1-0.500.51-0.200.20.40.6 0 0.500.51 0 0.5 0 0.5 0 0.5 1E γ =3.15 GeV E γ =3.85 GeV ρ E γ =9.6 GeVRe ρ E γ =4.5 GeV ρ -t’ [GeV ] III FIG. 5. Density matrix elements ρ λλ ′ at the G.-J. frame forthe ρ → ππ decay in the unpolarized process γp → ρ − ∆ ++ .Notations for the curves are the same as in Fig. 4. Data aretaken from Refs. [1, 2]. γ [GeV]-0.4-0.200.20.40 0.5 1 1.5 2-t [GeV ]-0.4-0.200.20.4 Σ (a) ( θ = 60 ο ) (E γ = 4 GeV) Model II(E γ = 3.75 GeV) Σ ( θ = 60 ο ) (1.01, -0.41)(0,0) Model I κ ∆ = 0κ ∆ = 4.34 (b)(d)(c) Σ Σ
FIG. 6. Sensitivity of photon polarization asymmetry Σ tomagnetic moments of ∆ in the Model I and ρ in the ModelII. Σ with and without κ ∆ are presented in (a) and (b) withrespect to − t and E γ . The raw (c) and (d) are the case of the ρ with and without κ ρ and λ ρ . forward-scattering region due to the nondiffractive π + ρ exchanges in the t -channel. The steep decrease of the to-tal cross section is well reproduced by the dominance ofthe π exchange over the ρ , and the canonical phase thusassigned to the ρ exchange describes the data with thedip-pattern in the differential cross section as well as theoscillatory behavior of the density matrix elements.Therefore, both the Model I and II can be appreciatedto yield the results valid up to E γ ≃
10 GeV. In particu-lar, the former model has the advantage of investigatingthe EM multipole moments of ∆-baryon in the low en-ergy region below E γ ≃ ACKNOWLEDGMENT
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