Decay angular distributions of K^* and D^* vector mesons in pion-nucleon scattering
aa r X i v : . [ nu c l - t h ] F e b Decay angular distributions of K ∗ and D ∗ vector mesons in pion-nucleon scattering Sang-Ho Kim, ∗ Yongseok Oh,
2, 1, † and Alexander I. Titov ‡ Asia Pacific Center for Theoretical Physics, Pohang, Gyeongbuk 37673, Korea Department of Physics, Kyungpook National University, Daegu 41566, Korea Bogoliubov Laboratory of Theoretical Physics, JINR, Dubna 141980, Russia
The production mechanisms of open strangeness ( K ∗ ) and open charm ( D ∗ ) vector mesons in π − p scattering, namely, π − + p → K ∗ + Λ and π − + p → D ∗− + Λ + c , are investigated withinthe modified quark-gluon string model. In order to identify the major reaction mechanisms, weconsider the subsequent decays of the produced vector mesons into two pseudoscalar mesons, i.e., K ∗ → K + π and D ∗ → D + π . We found that the decay distributions and density matrix elementsare sensitive to the production mechanisms and can be used to disentangle the vector trajectoryand pseudoscalar trajectory exchange models. Our results for K ∗ production are compared with thecurrently available experimental data and the predictions for D ∗ production processes are presentedas well. Our predictions can be tested at the present or planned experimental facilities. PACS numbers: 13.85.-t, 12.40.Nn, 13.75.Gx, 13.25.-k
I. INTRODUCTION
Investigation of open charm and open strangeness pro-duction processes is one of major hadron physics pro-grams at current or planned accelerator facilities whichare supposed to provide pion beams [1] or antiprotonbeams [2]. These facilities are expected to produce high-quality beams at energies high enough to produce strangeor charm hadrons. Understanding the dynamics of charmand strange quarks is an interesting topic as their massscale is between the light quark sector which is domi-nated by chiral symmetry and the heavy quark sectorwhere heavy quark spin symmetry takes a crucial role.Therefore, they are interpolating the two extreme regionsand the deviations or corrections to chiral symmetry andheavy quark symmetry are crucial to understand the un-derlying dynamics of the strong interaction.The heavy mass of the charm quark, in particular,leads to a very rich hadron spectrum with open or hid-den charm flavor, which includes exotic states that werenot observed in the light quark sector. Therefore, charmhadron spectroscopy is expected to open a new opportu-nity for unraveling the strong interaction. Besides, manyinteresting ideas using charm flavor have been suggested,which include the utilization of charm particles as a probeof nuclear medium at maximum compression, the studyof the properties of exotic
XY Z mesons and so on [5, 6].One of the important issues which are not fully re-solved at present is the charm/strangeness productionmechanism in hadron reactions. Since the reaction en-ergy is not high enough to be treated asymptotically, the ∗ [email protected] † [email protected] ‡ [email protected] The details of the physics programs using these beams can befound, for example, in the websites of Japan Proton AcceleratorResearch Complex (J-PARC) [3] and Facility for Antiproton andIon Research in Europe (FAIR) [4]. widely-used models for heavy quark production based onperturbative Quantum Chromodynamics (QCD) are notapplicable, and an essential improvement by includingnonperturbative contributions is indispensable. In theperturbative QCD approaches, in fact, charm quarks areproduced through gluon fragmentations. In order to pro-duce charm quarks in peripheral collisions, however, suchgluons must have a large momentum (large x ∼ x . .
2) in-side a nucleon. As a result, this mechanism can hardlybe the major production mechanism for heavy flavor pro-duction at relatively low energies.Therefore, it is legitimate to rely on the approachesbased on non-perturbative QCD background for describ-ing peripheral reactions. In the present work, we adoptthe quark-gluon string model (QGSM) developed byKaidalov and collaborators in Refs. [7–9], which has beenapplied for the evaluation of cross sections of the exclu-sive Λ c production in pp and ¯ pp collisions [10–12] andin πp collisions [13–15]. A novel feature of this model isthat the invariant amplitude of the binary reaction hasa form of the Regge amplitude, where the parameters ofan effective “Reggeons” are determined by unitary con-dition and additivity of intercepts and of inverse slopesof the Regge trajectories.For such an effective Reggeons or effective meson-trajectories exchanges, usually only the vector meson ex-change is considered. Since the intercept of vector mesontrajectory is larger than that of the corresponding pseu-doscalar trajectory with the same flavor quantum num-ber, the exchange of pseudoscalar meson trajectory isexpected to be suppressed. Therefore, this model wouldbe justified at large center-of-momentum energy squared s and small magnitude of the squared momentum trans-fer | t | . However, in order to fully understand the pro-duction mechanisms, more physical quantities should beexamined other than cross sections. In particular, suchphysical quantities should be sensitive to the productionmechanisms whose contribution to cross sections is rela-tively small. In fact, as we shall see below, the availabledata for K ∗ production suggest that the vector mesontrajectory exchange model needs to be modified to someextent.In the present work, we elaborate on the angular dis-tributions of pseudoscalar mesons originated from the de-cays of vector mesons produced in πN collisions. Morespecifically, we consider the production of K ∗ and D ∗ vector mesons, which decay into Kπ and Dπ , respec-tively. Therefore, the processes under consideration inthe present work are the two-step reactions of πN → K ∗ Λ → ( Kπ )Λ and πN → D ∗ Λ c → ( Dπ )Λ c , wherewe will specifically work on π − p collisions. In particu-lar, we focus on the angular distributions of K and D mesons produced by these reactions, which bear the in-formation on the production mechanisms of K ∗ and D ∗ vector mesons.This paper is organized as follows. In Sec. II we de-scribe QGSM, which will be used to describe K ∗ and D ∗ vector mesons. All the theoretical tools to investigatethe angular distributions of K and D mesons producedby the decays of the corresponding vector mesons are de-tailed as well. Then, in Sec. III, we show the results oncross sections, spin-density matrix elements, and decayangular distributions of vector mesons produced in π − p collisions. We summarize and conclude in Sec. IV. II. THE MODEL
The reactions under consideration in the present workare π − + p → V + Y → ( P + π )+ Y , where Y , V , P are fla-vored baryon, vector meson, and pseudoscalar meson, re-spectively. In the strangeness sector, Y = Λ(1116 , / + ), V = K ∗ (892 , − ), and P = K (494 , − ), while, in charmsector, Y = Λ c (2286 , / + ), V = D ∗ (2010 , − ), and P = D (1870 , − ) [16].The corresponding cross section for (2-body → dσ = (cid:18) πλ i | T fi | dt (cid:19) × (cid:18) k f d Ω f dM V π (cid:19) , (1)where T fi is the invariant amplitude for the productionprocess and λ i ≡ λ ( M π , M N , s ) is the K¨all´en functiondefined as λ ( x, y, z ) ≡ x + y + z − xy − yz − zx .Here, M π and M N stand for the pion mass and the nu-cleon mass, respectively, and we use M V for the vectormeson mass. The Mandelstam variables for the produc-tion process are defined as s = ( p π + p p ) = ( p V + p Y ) and t = ( p p − p Y ) = ( p π − p V ) , where p π , p p , p V , and p Y are the four momenta of the pion, proton, produced(virtual) vector meson, and hyperon, respectively. Thesolid angle and the magnitude of the three momentumof outgoing pseudoscalar meson in the rest frame of thevector meson are represented by Ω f and k f , respectively.The averaging over the initial spin states and sum overthe final spin states is understood as well. The invariant amplitude can be expressed as T fi = A m f ,λ V ; m i p V − M + iM Γ tot D λ V (Ω f ) , (2)where m i and m f denote the spin projections of incom-ing and outgoing baryons, respectively, and λ V representsthe spin projection of the produced virtual vector meson. M and Γ tot are the pole mass and the total decay widthof the produced vector meson, respectively. The ampli-tudes of the π − + p → V + Y and V → P + π reactionsare denoted by A and D , respectively. The decay processof vector mesons is considered in its rest frame. In thiscase, the amplitude of the vector meson decay into twopseudoscalar mesons has the simple form of D λ = 2 c r π Y λ (Ω f ) , (3)where the constant c is related to the V → P + π decaywidth Γ f as c = 6 πM V Γ f k f , (4)with k f being the magnitude of the three momentum ofthe final-state particles in the rest frame of the vectormeson. Integration of dσ in Eq. (1) over dM V and d Ω f leads to the well-known result for the corresponding un-polarized cross section, dσdt = Br16 πλ i |A fi | , (5)with Br = Γ f / Γ tot when Γ tot ≪ M V .Recent studies of strangeness and charm productionat a few dozen GeV show that this cross section can besuccessfully evaluated in the framework of QGSM sug-gested by Kaidalov [7, 8] and later developed and re-fined in a number of theoretical works developed, for ex-ample, in Refs. [9–15]. QGSM is based on the planarquark diagram decomposition and unitary conditions [8],and it allows to represent the amplitude of the binary a + b → c + d reaction in terms of an effective Reggeamplitude, where the effective trajectory α R ( t ) and theenergy scale parameter s ab ; cd are determined by the well-established parameters of the elastic a + b → a + b and c + d → c + d reactions using so-called the planar dia-gram decomposition. An example of the planar diagramdecomposition is depicted in Fig. 1 for the reaction of π − + p → D ∗− + Λ + c , where it is assumed that the ampli-tude is dominated by the effective D ∗ trajectory with pa-rameters completely determined by the non-linear ρ and J/ψ meson trajectories as found from the meson spec-troscopy studies [8, 17, 18]. Similarly, one can write theplanar diagram decomposition for the K ∗ Λ productionwith substitution of the
J/ψ trajectory by the φ mesontrajectory. We refer the details to Ref. [11].Diagrammatic representations of the effective π − + p → K ∗ + Λ and π − + p → D ∗− + Λ + c reactions are shown in p Λ + c π − D ∗− d ¯ uuud d ¯ ccud π − d ¯ uuud p uud p d ¯ u π − D ∗ × D ∗− Λ + c d ¯ ccud cud ¯ cd D ∗− Λ + c ρ J/ Ψ FIG. 1. Planar diagram decomposition for the reaction of π − + p → D ∗− + Λ + c . p Λ π − K ∗ d ¯ uuud d ¯ ssud π − d ¯ uuud p cud Λ + cd ¯ c D ∗− K ∗ D ∗ FIG. 2. Diagrammatic representation of the effective π − + p → K ∗ + Λ and π − + p → D ∗− + Λ + c reactions. Fig. 2. The corresponding spin-independent amplitudesread A Vfi = g s ¯ s Γ ( − α V R ( t ) ) (cid:18) ss V R (cid:19) α V R ( t ) − (6)with α V R p Λ ( t ) = 0 .
414 + 0 . t , s V R p Λ = 1 .
66 GeV ,¯ s p Λ = 1 GeV for π − + p → K ∗ + Λ, and α V R p Λ c ( t ) = − .
02 + 0 . t , s V R p Λ c = 4 .
75 GeV , ¯ s p Λ c = 1 GeV for π − + p → D ∗− + Λ + c . The trajectories of ρ , φ , and J/ψ , as well as the energy-scale parameters s V R are de-termined following the prescription described in Ref. [11].The residual factor g will be determined in the nextSection by comparison with the available experimentaldata for the π − + p → K ∗ + Λ reaction, which leads to g / π ≃ . K ∗ → K + π and D ∗ → D + π strongly depend on the spin of the partic-ipating particles, the spin structure of the reaction am-plitudes of Eq. (6) should be specified. This, in fact, isthe key component which can distinguish different pro-duction mechanisms. It can be done by “dressing” thespin-independent amplitude by the spin-factor S fi whichcarries the symmetry of exchanged Reggeon [11], i.e., A fi → A m f ,λ V ; m i = A fi N S m f ,λ V ; m i , (7)with the normalization factor N = X m f ,m i ,λ V (cid:12)(cid:12)(cid:12) S m f ,λ V ; m i (cid:12)(cid:12)(cid:12) . (8)The K ∗ meson couplings in the spin-factor S fi reads S m f ,λ V ; m i = ǫ µναβ q µ p V α ε ∗ β ( λ V ) × ¯ u m f (Λ) (cid:20) (1 + κ K ∗ p Λ ) γ ν − κ K ∗ p Λ ( p p + p Λ ) ν M p + M Λ (cid:21) u m i ( p ) , (9)where q = p V − p π = p p − p Λ is the momentum transferand κ K ∗ p Λ = 2 .
79 is the tensor coupling constant ob-tained from the average value of the Nijmegen soft-corepotential [19, 20]. The Dirac spinors of initial baryon andfinal baryon are denoted by u m i and u m f , respectively,and ε ( λ V ) is the polarization vector of the produced vec-tor meson. Generalization to the case of charm produc-tion may be achieved by the substitution M Λ → M Λ c , M K ∗ → M D ∗ , and so on. Because of the lack of infor- mation, we assume κ K ∗ p Λ = κ D ∗ p Λ c as in Ref. [21]. Thenormalization factor N in Eq. (7) is introduced to com-pensate for the artificial s and t dependence generatedby S fi .The differential cross section is then written as dσdt d Ω f = dσdt W (Ω f ) , (10)where W (Ω f ) = X m i ,m f ,λ V ,λ ′ V M m f ,λ V ; m i M ∗ m f ,λ ′ V ; m i × Y λ V (Ω f ) Y ∗ λ ′ V (Ω f ) , (11)with M m f ,λ V ; m i = 1 N S m f ,λ V ; m i . (12)For definiteness with isospin quantum number we con-sider K ∗ → K + π − and D ∗− → D − π decays. As is wellknown, since the decay angular distribution of outgoing K + or D − is analyzed in the virtual vector meson’s restframe, there is an ambiguity in choosing the quantizationaxis. One may choose the quantization axis anti-parallelto the outgoing hyperon Y in the center-of-momentumframe of the production process. Or the quantizationaxis may be defined to be parallel to the incoming pion,i.e., the initial beam direction. Following the conventionof Ref. [22], the former is called the s -frame and the latterthe t -frame. The decay probabilities are expressed in terms of thespin-density matrix elements ρ λλ ′ , where λ V is abbrevi-ated as λ , which are determined by the amplitudes ofEq. (12). Depending on the polarization state of the ini-tial and final states, we are interested in the followingtwo cases:1. Unpolarized case, where the spin-density matrix isgiven by ρ λλ ′ = X m i = ± , m f = ± M m f ,λ ; m i M ∗ m f ,λ ′ ; m i , (13)2. Recoil polarization case, when the spin of the out-going hyperon ( Y ) is determined by their decay dis-tribution using that it is self-analyzing. Then, de-pending on the spin state of the hyperon, we havetwo kinds of spin-density matrices defined as ρ ± λλ ′ = X m i = ± M m f ,λ ; m i M ∗ m f ,λ ′ ; m i . (14)Here, ρ + and ρ − correspond to the cases when thespin or helicity of the produced hyperon is m f =+ and − , respectively.Denoting the polar and the azimuthal angles of theoutgoing pseudoscalar K (or D ) mesons by Θ and Φ, In the case of vector meson photoproduction, the former is calledthe helicity frame, while the latter corresponds to the Gottfried-Jackson frame [23]. respectively, the decay angular distributions can be ex-pressed in terms of the spin-density matrix elements as W (Ω f ) = 34 π h ρ cos Θ + ρ sin Θ − ρ − sin Θ cos 2Φ − √ ρ ) sin 2Θ cos Φ i , (15)for the unpolarized case and W ± (Ω f ) = 34 π (cid:20) ρ ± cos Θ + 12 (cid:0) ρ ± + ρ ±− − (cid:1) sin Θ − ρ ± − sin Θ cos 2Φ − √ (cid:0) ρ ± − ρ ±− (cid:1) sin 2Θ cos Φ (cid:21) , (16)for the case of recoil polarization. Here, we made use ofthe Hermitian conditions: ρ − = ρ − , ρ = ρ , and ρ − = ρ − . In addition, for unpolarized reaction, wealso have the sum rule ρ + ρ + ρ − − = 1 and thesymmetry conditions, ρ = ρ − − and ρ = − ρ − . Inthe case of recoil polarization, however, these additionalrelations do not hold.As was mentioned earlier, the purpose of the presentwork is to test the validity of the dominance of vectormeson trajectory exchange. This assumption is based onthe observation that the intercept of the K ∗ ( D ∗ ) vectormeson, for instance, is larger than that of the correspond-ing pseudoscalar K ( D ) meson trajectory [17]. However,other mechanisms cannot be excluded, and the contribu-tion from such mechanisms should be verified by physicalquantities related to the spin structure of the productionmechanisms. In fact, as we will see later, the availabledata for density matrix elements suggest that there ex-ist contributions from mechanisms other than vector tra-jectory exchange. Therefore, in addition to vector tra-jectory exchanges, we consider the exchanges of effectivepseudoscalar K and D trajectories. In this case, the spin-independent amplitude reads A P Sfi ≃ g Γ ( − α PS R ( t ) ) (cid:18) ss PS0 R (cid:19) α PS R ( t ) (17)with α PS R p Λ ( t ) = − .
151 + 0 . t , α PS R p Λ c ( t ) = − .
611 +0 . t [18]. The energy scale parameters determinedby the flavor content of the vertices are assumed to bethe same as in the vector meson exchange case so that s PS0 R p Λ = s V R p Λ and s PS0 R p Λ c = s V R p Λ c . The spin factor S fi now reads S PS m f ,λ V ; m i = ε ∗ µ ( λ V ) q µ ¯ u m f (Λ) γ u m i ( p ) . (18) III. RESULTS AND DISCUSSION
In this Section, we present numerical results on differ-ential cross sections, spin-density matrix elements, anddecay angular distributions of K and D mesons in πN scattering. t max − t (GeV ) −1 d σ / d t ( µ b / G e V ) π − p → K *0 Λ p π = 6 GeV/ c (a) t max − t (GeV ) −2 −1 d σ / d t ( µ b / G e V ) π − p → D * − Λ c + p π = 15 GeV/ c (b) FIG. 3. Unpolarized differential cross sections of (a) π − + p → K ∗ + Λ and (b) π − + p → D ∗− + Λ + c for the vector (solidcurves) and pseudoscalar (dashed curves) Reggeon exchanges. The experimental data for π − + p → K ∗ + Λ are from Ref. [22]. A. Unpolarized cross sections
By collecting all information, the unpolarized differ-ential cross sections of the π − + p → K ∗ + Λ and π − + p → D ∗− + Λ c reactions for the vector (V) andpseudoscalar (PS) effective Reggeon exchanges are writ-ten as dσ (V) dt = πλ i (cid:16) s ¯ s (cid:17) (cid:20) ( g V0 ) π (cid:21) (cid:2) Γ ( − α V ( t ) ) (cid:3) × (cid:18) ss R V (cid:19) α V ( t ) − ,dσ (PS) dt = πλ i (cid:20) ( g PS0 ) π (cid:21) (cid:2) Γ ( − α PS ( t ) ) (cid:3) (cid:18) ss R PS (cid:19) α PS ( t ) . (19)The residual factor g is, in general, a function of t , andshould be determined by the comparison with experi-mental data. We use ( g V0 ) / π = 0 .
796 for the vector me-son trajectory exchange, which is found from comparisonwith the available experimental data for K ∗ production.We use this value for both the strangeness and charmproduction processes as we do not have any data forcharm vector meson production. Since we are interestedin identifying the major production mechanisms, we needto be able to distinguish between the pseudoscalar me-son trajectory exchange and the vector meson trajectoryexchange through measurable physical quantities. Sincethe pseudoscalar exchange mechanism is expected to besmall, we consider two extreme cases, namely, vector-exchange dominance and pseudoscalar-exchange domi-nance. For this purpose, we adjust the value of g PS0 toachieve the condition that dσ (PS) /dt = dσ (V) /dt at zerovector meson production angle, i.e., at t = t max . Thisleads to ( g PS0 ) / π = 1 . K ∗ and D ∗ mesons production, respectively. Of course, the realisticcase is between these two extreme cases, and the relativestrength of the two mechanisms should be determined byexperimental data.The obtained differential cross sections for K ∗ and D ∗ production are exhibited in Figs. 3(a) and (b), re-spectively. Throughout the present study, the initialpion momentum in the laboratory frame is chosen to be p π = 6 GeV/ c for strangeness production and 15 GeV/ c for charm production. The vector (V) and pseudoscalar(PS) Reggeon exchanges are shown by the solid anddashed curves, respectively, together with available ex-perimental data of Ref. [22] for K ∗ production. Althoughthe energy scale is different, it turns out that the crosssection of charm production is suppressed compared withstrangeness production, which is consistent with the ob-servation made in Ref. [14]. One can see that both thevector-type Reggeon exchange and the pseudoscalar-typeReggeon exchange exhibit a similar t -dependence in dif-ferential cross sections. This resemblance is clearly seenin the case of charm production, although the availabledata seem to prefer the vector-type exchange in the caseof strangeness production. Therefore, the t -dependenceof cross sections cannot clearly distinguish the two ex-changes. As we will see in the next subsections, however,the situation changes for spin-density matrix elementsand the angular distributions of K ∗ → Kπ and D ∗ → Dπ decays, where the difference between the two types of ex-changes is revealed even at the qualitative level. B. Spin-density matrix elements
The results for spin-density matrix elements ρ λλ ′ de-fined in Eq. (13) are presented in Fig. 4 for K ∗ and D ∗− production as functions of ( t max − t ). We alsolimit our consideration to relatively small values of | t | such that | t max − t | ≤ , where the applicability ofQGSM can be justified. Shown in Fig. 4 are the resultsfor the vector-type Reggeon exchange model and for thepseudoscalar-type Reggeon exchange model, which arecalculated in the s - and t -frames. Our results numer-ically confirm the symmetry properties, ρ = ρ − − , ρ ± = ρ ± , ρ ± = − ρ ∓ , and ρ − = ρ − .In the case of vector-type Reggeon exchange, the ma-trix elements ρ λλ ′ with | λ | = | λ ′ | = 1 are enhanced. This −0.200.20.40.6 ρ λλ ′ s- frame −0.200.20.40.6 t- frame −0.400.40.81.2 ρ ρ Re ρ ρ Re ρ s -frame −0.400.40.81.2 ρ λλ ′ t -frame −0.200.20.40.6 ρ λλ ′ s -frame −0.200.20.40.6 t -frame −0.400.40.81.2 s -frame −0.400.40.81.2 ρ λλ ′ t -frame π − p → K ∗0 Λπ − p → D ∗− Λ c+ V exchange t max − t (GeV ) PS exchange t max − t (GeV ) t max − t (GeV ) (a) (b) (c)(e) (f) (g) (h)(d) t max − t (GeV ) FIG. 4. The spin-density matrix elements ρ λλ ′ defined in Eq. (13) as functions of ( t max − t ) (a)–(d) for K ∗− production at p π = 6 GeV/ c and (e)–(f) for D ∗− production at p π = 15 GeV/ c . The results for vector meson (V) and pseudoscalar (PS)Reggeon exchanges are in (a), (b), (e), (f) and (c), (d), (g), (h) panels, respectively. The results in (a), (c), (e), (g) are obtainedin the s -frame, while those in (b), (d), (f), (h) are in the t -frame. ascribes to the spin structure ǫ µναβ q µ p V α ε ∗ β ( λ V ) of theamplitude in Eq. (9). In the vector meson rest frame,where p V = ( M V , , ,
0) and q = − p π , this factor is pro-portional to the vector product of ε ∗ ( λ V ) × p π . In the s -frame and small momentum transfers, p π has a large z component and a small x component, which leads to ε ∗ ( λ V ) × p π ≃ iλ V ε ∗ ( λ V ) | p π | and thus causes the largeenhancement of ρ | λ | =1 , | λ ′ | =1 . In the t -frame, p π is par-allel to the quantization axis, and this leads to that ρ λλ ′ with either λ = 0 or λ ′ = 0 vanish. We also note that ρ − = 0 at t = t max . This is because of the relation ρ − ∝ sin θ , where θ is the scattering angle of the vec-tor meson in the c.m. frame for the scattering process.So is the matrix element ρ +1 − as seen in Fig. 5.In the case of pseudoscalar-type Reggeon exchange,however, the situation is quite different. The produc-tion amplitude of this mechanism is proportional to thescalar product, ε ∗ ( λ V ) · p π , which leads to a strong en-hancement of ρ in the t -frame, so that ρ = 1 and allthe other ρ λλ ′ vanish.Shown in Fig. 5 are the results for ρ + λλ ′ defined in Eq. (14). In this case, the spin alignment of the outgoinghyperon is fixed to be m f = + . The absolute values of ρ ± are smaller than those of ρ by about a factor of twobecause of the difference in the numerators in Eqs. (13)and (14). The spin-density matrix elements ρ − λλ ′ can beobtained from ρ + λλ ′ using the symmetry relations [23]: ρ − = ρ + − − , ρ − = ρ +00 , ρ −− = ρ + − , ρ − = − ρ +0 − , andso on.In Figs. 6 and 7, we compare our results with the avail-able experimental data of Ref. [22] for K ∗ productionat s - and t -frames, respectively. Although the vector-exchange mechanism leads to a better agreement withthe data than the pseudoscalar-exchange model, we cansee that the vector-exchange model alone cannot success-fully explain the data. New experimental data for K ∗ production with higher precision are, therefore, stronglydesired. In D ∗ production, the difference is also large We could confirm this conclusion through the comparison withthe data obtained at p π = 4 . −0.200.20.4 ρ + λλ ′ s -frame −0.200.20.4 t -frame −0.400.40.8 ρ +00 ρ +11 ρ +10 ρ +1−1 ρ +−10 ρ +−1−1 s -frame −0.400.40.8 ρ + λλ ′ t -frame −0.200.20.4 ρ + λλ ′ s -frame −0.200.20.4 t -frame −0.400.40.8 s -frame −0.400.40.8 ρ + λλ ′ t -frame π − p → K ∗0 Λπ − p → D ∗− Λ c+ (a) (b) (c) (d)(e) (f) (g) (h) t max − t (GeV ) V exchange PS exchange t max − t (GeV ) t max − t (GeV ) t max − t (GeV ) FIG. 5. The same as in Fig. 4 but for ρ + λλ ′ . enough to be verified by experiments and the analysescan be done at current or future experimental facilities. C. Angular distributions of vector meson decays
The polar angle distributions of outgoing K and D mesons are obtained by integrating W (Θ , Φ) of Eqs. (15)and (16) over the azimuthal angle Φ, which gives23 W (Θ) = ρ cos Θ + ρ sin Θ , W ± (Θ) = ρ ± cos Θ + 12 (cid:0) ρ ± + ρ ±− − (cid:1) sin Θ . (20)These distributions are presented in Fig. 8 for the pro-duction and decays of K ∗ and D ∗ mesons at | t max − t | =0 . with p π = 6 and 15 GeV/ c , respectively.In all cases, one can observe maxima at Θ = π for thevector trajectory exchange while minima are observed atthe same angle for the pseudoscalar trajectory exchange.In other words, the distribution functions for the vec-tor trajectory exchange display a cosine function shape,while those of the pseudoscalar trajectory exchange show a sine function shape. This is a direct consequence of thespin density matrix elements ρ and ρ shown in Figs. 4and 5.The azimuthal angle distributions at a fixed polar an-gle Θ can also be obtained from Eqs. (15) and (16). AtΘ = π , we have4 π W (Θ = π , Φ) = ρ − ρ − cos 2Φ , π W ± (Θ = π , Φ) = 12 (cid:0) ρ ± + ρ ±− − (cid:1) − ρ ± − cos 2Φ . (21)The corresponding distributions are shown in Fig. 9 at | t max − t | = 0 . . In the s -frame, the matrix el-ement ρ − takes a positive value for vector-type ex-change and a negative value for pseudoscalar-type ex-change. This difference makes that W ( π , Φ) of vector-type exchange and pseudoscalar-type exchange are outof phase. The amplitudes of the oscillations in W arefound to be larger than those of W ± , which reflects thedifferences in ρ − as shown in Figs. 4 and 5. For thepseudoscalar-Reggeon exchange in the t -frame, ρ , + λ,λ ′ = 0for | λ | = | λ ′ | = 1, and, therefore, the corresponding dis- t max − t (GeV ) −0.40.00.40.8 ρ t max − t (GeV ) −0.40.00.40.8 R e ρ π − p → K ∗0 Λ ( s -frame) t max − t (GeV ) −0.40.00.40.8 ρ − PSV (a) (b) (c)
FIG. 6. Spin-density matrix elements for K ∗ production in the s -frame. The panels (a), (b), and (c) correspond to ρ ,Re ρ , and ρ − matrix elements, respectively. The vector and pseudoscalar Reggeon exchange models are depicted by thesolid and dashed curves, respectively. The experimental data are from [22]. t max − t (GeV ) −0.40.00.40.8 ρ t max − t (GeV ) −0.40.00.40.8 R e ρ π − p → K ∗0 Λ ( t -frame) t max − t (GeV ) −0.40.00.40.8 ρ − PSV (a) (b) (c)
FIG. 7. The same as in Fig. 6 but in the t -frame. tributions W ( π , Φ) vanish identically.
IV. SUMMARY AND CONCLUSION
In summary, we investigated the reactions of openstrangeness K ∗ and open charm D ∗ vector mesons in πN scattering based on the quark-gluon string model.We found that unpolarized cross sections of K ∗ mesonproduction is satisfactorily described by QGSM with vec-tor trajectory exchange. Although the contribution frompseudoscalar trajectory exchange is expected to be small,it also gives a similar t -dependence of differential crosssections as the vector-type exchange model. Therefore,differential cross sections cannot be used to disentanglethe two production mechanisms.In order to verify the mechanisms of vector meson pro-duction, we then studied the angular distributions of vec- tor meson decays. Unlike the cross sections, the spindensity matrix elements are sensitive to the spin struc-ture of the production amplitude and, as a result, theyshow very different t -dependence and can be used to dis-tinguish the vector and pseudoscalar exchanges. Fur-thermore, the density distribution functions are found tohave complete different angle-dependence depending onthe production mechanisms and can be used to probethe spin structure of the reaction amplitudes. In fact,the available data for spin density matrix elements of K ∗ production show that the major production mecha-nism would be the vector-type exchange but it requires anoticeable contributions from the pseudoscalar-type ex-change. Because of the limited experimental data, wecannot estimate the relative strength between the vectorand pseudoscalar exchanges, and, therefore, new data arestrongly called for to investigate strangeness and charmproduction mechanisms. ( / ) W ( Θ ) s -frame t -frame ( / ) W + ( Θ ) s -frame t -frame ( / ) W ( Θ ) s -frame t -frame ( / ) W + ( Θ ) s -frame t -frame π − p → K ∗0 Λ π − p → K ∗0 Λπ − p → D ∗− Λ c + π − p → D ∗− Λ c + PSV (a) (b) (c) (d)(h)(g)(f)(e)
FIG. 8. Angular distributions W (Θ) of Eq. (20) for K ∗ and D ∗ excitations are shown in the upper and lower panels,respectively. The panels (a), (b), (e), (f) are for W (Θ) and (c), (d), (g), (h) are for W + (Θ). The results are given atboth the s - and t -frames. The vector and pseudoscalar Reggeon exchange cases are depicted by the solid and dashed curves,respectively. Calculation is done for | t max − t | = 0 . at p π = 6 GeV/ c for K ∗ production and 15 GeV/ c for D ∗ production. We also presented our predictions for charm produc-tion. In this case, the t -dependence of differential crosssections of the vector and pseudoscalar exchanges is evencloser to each other because of the similarity in the slopeof vector and pseudoscalar trajectories, and thus the mea-surements of differential cross sections does not help pindown the production mechanisms. However, spin densitymatrix elements and decay angular distributions are verysensitive to the production mechanisms as in the case of K ∗ production, and more detailed studies on these quan-tities are expected to shed light on our understanding ofthe strong interaction. In particular, the measurementsfor K ∗ and D ∗ productions are complementary to each other and would be important to understand the depen-dence of the production mechanisms on the quark massscale. All these predictions can be tested and verifiedin future experimental programs with pion beams, forinstance, at J-PARC facility. ACKNOWLEDGMENTS
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