Pion induced Reactions for Charmed Baryons

We study pion induced reactions for charmed baryons $B$, $\pi$ + N $\rightarrow$ $D^*$ + $B$. First we estimate charm production rates in comparison with strangeness production using a Regge model which is dominated by vector ($D^*$ or $K^*$ ) Reggeon exchange. Then we examine the production rates of various charmed baryons $B$ in a quark-diquark model. We find that the production of excited states are not necessarily suppressed, a sharp contrast to strangeness production, which is a unique feature of the charm production with a large momentum transfer.


I. INTRODUCTION
Observations of new hadrons have been stimulating diverse activities in hadron physics, see for instance, Ref.
[1]. Evidences first observed at electron facilities such as KEK, SLAC and BES [2][3][4][5] are now receiving strong support from recent LHCb experiments [6,7]. Many new hadrons have been found near the threshold regions of charm or bottom quarks. Intuitively, excited heavy quarks break a string followed by a creation of a light quark-antiquark pair, forming the exotic hadrons with multiquarks near the threshold. To understand the features of the new findings, therefore, requires systematic studies of the dynamics from light to heavy quark regions.
So far, many of the new observations were made for mesons. In contrast, progress for baryons has not been achieved much. In fact, the number of known heavy quark baryons is much less than that of light quark baryons. The study of charmed baryons is important not only for heavy but also for light quark dynamics, which in turn will be linked to the physics of the new hadrons and eventually to the unsolved problems of QCD.
Under the above background, an experimental proposal is being made for the new pion beam facility at J-PARC [8]. The expected pion energy will reach over 20 GeV in the laboratory frame which is sufficient to excite charmed baryons up to around 1 GeV. This is a challenging experiment since there has been no experiment after the one at Brookhaven almost thirty years ago [9]. The relevant reaction has been chosen, i.e., where D * is the charmed vector meson and B a charmed baryon. The reason D * is selected in the reaction is due to experimental advantage as compared to the production of D meson. The purpose of this paper is to perform a theoretical study for the above reaction, while experimental feasibility is now under investigation. The study of such reactions is a challenging problem, because 1) not many studies have been performed so far, 2) production rates should reflect structure of charmed baryons, and furthermore 3) charm production mechanism from the threshold to a few GeV regions is not well understood.
The structure of charmed baryons have been studied in a quark model [10,11]. One of unique features due to the presence of a charm quark is the so-called isotope shift. In the light flavor sector where the three quarks have a similar mass, the two independent internal motions of ρ and λ modes are degenerate, which in the presence of a heavy quark split and appear differently in the spectrum. This seems to be the case already in the strange baryons, as seen in the inversion of the mass ordering in Σ(1775)-Λ(1830). It is then very important to perform systematic studies from the light to the heavy flavor sectors. This paper is organized as follows. In section 2, we estimate the rate of charm production using a Regge model in comparison with strangeness production. In section 3, we compute the production rates of various charmed baryons B, up to the orbital excitations of d-wave (l = 2) in a heavy quark-diquark description of B. The result indicates that the production of excited states B is not necessarily suppressed in comparison with strange hyperon production. In section 4, we discuss prospects and summarize the present work.

II. ESTIMATION OF CROSS SECTIONS
Let us consider forward angle scattering, where the t-channel dynamics as shown in Fig. 1 dominates, and the Regge model is expected to be a good prescription. Many experiments have shown that cross sections are of forward peak (diffractive) at energies beyond a few GeV, which is the region of charm production also. For strangeness production, a reaction relevant to the present study, π + p → K * + B s , was performed long ago [12,13]. They have shown clearly a forward peak structure, which indicates the t-channel mechanism in the forward angle region. In the Regge theory [14], the scattering amplitude is first expanded into partial waves in the t-channel scattering region (s < 0, t > 0), which is then analytically continued to the physical region of s-channel scattering (s > 0, t < 0). The sum over integer angular momentum l is then equivalently expressed by the Regge pole terms which are the residues of the scattering amplitude in the complex angular momentum plane. The pole is a function of t and is identified with a Regge trajectory α(t). The amplitude expressed by the Regge poles is then referred to as the Reggeon exchange amplitude.
The advantage of the Regge theory is that it determines the asymptotic behavior of the cross section of binary reactions, which describes well the observed s-dependence. Among various contributions of different trajectories (Reggeons), the dominant one is given by the one of the largest α(t). For example, the vector Reggeon is more dominant than the pseudoscalar Reggeon. For our present estimation, we employ the Kaidalov's prescription for the vector Reggeon exchange [15,16], Here p is the relative momentum of the initial state in the center of mass system ands a universal scale parameter. In the present study of ratios the parameters is not important. The other scale parameter s 0 depends on flavors of the Reggeon, and is determined by the probabilistic picture [15], s 0 (charm) = 4.75 GeV 2 , s 0 (strange) = 1.66 GeV 2 .
For the trajectories α V (t), we employ a non-linear parametrization where the parameters α 0 , γ and T are given in Ref. [17].
In this paper, we show the result of only the differential cross section of Eq. (3). One could also obtain the total cross section, but here we will not do it, because there is ambiguity in the form factor (t-dependence). In Eq. (3) we employ the one derived from the Regge's method which is analytically continued from the t-channel scattering region to the s-channal scattering region. This does not necessarily reproduce the observed t-dependence well. In fact, an alternative parametrization is possible when data are available [16,18,19]. Thus our strategy here is to investigate the forward cross section dσ/dt(θ = 0) for charm and strangeness productions, expecting that the Regge model works best in the forward angle region.
In Fig. 2, we show the results as functions of s/s th , where s th is the s-value at the threshold. Two curves are plotted in an arbitrary unit with keeping their ratio determined by Eq. (3). The ratio of the charm to strangeness production varies from 10 −3 near the threshold s/s th ∼ 1 to 10 −5 at large energies s/s th ∼ 10. The expected experiments at JPARC will be done most efficiently at s/s th ∼ 2, where the rate of charm production is smaller than strangeness production by a factor about 10 −4 . Therefore, if one uses the K * production cross sections of order 10 [µb] [12,13], the expected one for charm production is of order 1 [nb].

A. Quark-diquark baryons
In this section, baryons are described as two-body systems of a quark and a diquark. Charmed baryons are then composed of a heavy quark and a light diquark. The relative motion of the quark and diquark is described by the λ coordinate, one of the Jaccobi coordinates of a three-body system as shown in Fig. 3. The internal motion of the diquark as described by the other variable ρ is implicit in the quark-diquark model. Due to spin-spin interaction, the pair of 3 S ρ 0 quarks (d 0 ) is considered to have a lower mass than the pair of 3 S ρ 1 quarks (d 1 ) . In general, we can also consider internal excitations of diquarks. Furthermore, the λ and ρ modes can couple and mix. In this paper, however, we consider only λ motions of (orbitally) ground state diquarks of the above two kinds, d 0 and d 1 , because the reaction mechanism that we consider as shown in Fig. 1 (right) excites dominantly a λ mode. The quark-diquark wave functions of the λ modes are summarized in Appendix B. We have then made a tentative assignment of these states with the nominal ones listed in PDG when available [20] as shown in Table I. We have also made arbitrary assignment for the unknown states to fill the corresponding ones by simply guessing their masses. The latter are shown in Table I with a * symbol. As shown in Fig. 1 in the t-channel process, a charmed Reggeon is exchanged and couples with a quark in the initial nucleon transformed into a charm quark forming a charmed baryon in the final state. Our calculation here is performed under several assumptions.
• As in the previous section, we consider vector (V = D * or K * ) Reggeon exchanges because at high energies the V Reggeon dominates.
• The cross section shows a forward peak. Therefore, we compute the differential cross sections only at the forward angle.
• We focus on ratios of excited charmed baryon production as compared to ground state production.
The main issue in this section is the computation of various baryon matrix elements, which determines the production rates. For this purpose, we need a vertex for quark-diquark baryons. In addition, we also consider a meson vertex to evaluate the whole t-channel diagram. Thus we introduce the following two interaction Lagrangians, Here, f and g are coupling constants, and q and c denote the spinors of the light (q = u, d) and charm quarks, respectively.

B. Amplitudes
Let us first look at the matrix element of the πV V coupling of Eq. (6), where k π , k V and q are the momentum of the initial pion, of the final V and of the exchanged V meson, respectively. e α,β are the polarization vectors of either the final or the intermediate vector mesons. In these manipulations, we selected the dominant term assuming that the reaction energy is not relativistically too large as in the case for s/s 0 < ∼ 2.
Next, we compute the baryon matrix element of L V qc , where ϕ i,f are the two component spinors for the initial light quark and the final charm quark, respectively. To proceed, we pick up only terms that contain the spatial component of the V meson, because when this V meson is contracted with another from the πV V vertex, only the spatial component survives as Eq. (8) implies. Hence we find Now combining the matrix elements Eqs. (8) and (10), we can write down the scattering amplitude as where is the Reggeon propagator, and J f i the baryon transition current, Here we have defined the effective momentum transfer which takes into account the recoil of the center of mass motion due to the change in the masses of q and c quarks [21].
To further simplify the computation, the quark momenta p i and p f are approximated to take a fraction of the baryon momentum, Note that for the initial state the pion momentum (and hence the nucleon momentum) is sufficiently large such that the mass of the light quarks in the nucleon is neglected. Now for forward scattering where all momenta are collinear along the z-axis, only the spin current term survives in the scattering amplitude: where the constant factors which are irrelevant when taking ratios of the production rates are ignored. The polarization of V can be either longitudinal (z) or transverse (x, y), but the longitudinal contribution vanishes. Moreover, for the transverse polarization, the first term vanishes. Finally, we obtain a rather concise formula for the amplitude Here e ⊥ denotes the transverse vector, and hence the transverse spin induces the transition, as expected for the vector (J P = 1 − ) exchange process.

C. Production rates
We have computed the transition amplitudes t f i from the nucleon i ∼ N to various charmed baryons f ∼ B. For charmed baryons, we consider all possible states including the ground, p-wave and d-wave excitations. The production rates are computed by Using the results of the amplitudes as shown in Appendix A, we find In these expressions, C is the geometric factor of the matrix element f | e ⊥ · σ e i q ef f · x |i determined by the spin, angular momentum and total spin of the baryon, while I L (L = 0, 1, 2) contains dynamical information of the baryon wave function. K is the kinematic factor and γ the following isospin overlap factor γ = 1 √ 2 for Λ baryons , By using the baryon wave functions as summarized in Appendix B and C, the geometric factors C and the production rates R are computed. In Table I, results are shown for both charm and strangeness productions at the pion momentum in the laboratory frame, k Lab π = 20 GeV for charm production and k Lab π = 4.2 GeV for strangeness production. These momenta correspond to s/s th = (see text for assignment), spin-dependent coefficients C and the ratios of production rates R given in Eq. (19). The second and third rows are the ratios R for the strange and charmed baryons, respectively, which are normalized to the ground state Λ. They are computed at k Lab π = 4.2 GeV for the strange, and at k Lab π = 20 GeV for the charmed baryons. Herein below we make several observations.
• In general the production rates for Λ baryons are larger than for Σ baryons. This is a consequence of SU(6) symmetry of the quark-diquark baryons.
• Some excited Λ c states with a higher l have a similar or even larger production rate than the ground state, in particular Λ c (1/2 − ) and Λ c (3/2 − ), and Λ c (3/2 + ) and Λ c (5/2 + ). This is due to large overlap of the wave functions when the momentum transfer is large, typically around 1 GeV for charm production. The momentum transfer value together with the size of the baryons determines an optimal angular momentum transfer ∆l. For charm production this occurs at around ∆l ∼ 1, while for strangeness production at ∆l << 1. Mathematically, this is explained by the combination of the power term (q ef f /A) l and the form factor exp(−(q ef f /2A) 2 ) as in Eqs. (A13) and (A16). In hypernucleus production, the same mechanism has been well appreciated, demonstrating the success in the studies of reaction and structure [21].
• The above pairs of Λ's form a spin-orbit (LS) doublet in the quark model, or in the heavy quark limit the heavy quark doublet [22]. Their relative production rates are then determined in a model-independent manner up to a kinematic factor.
• We can similarly compute the amplitude for P (pseudoscalar)-Reggeon exchanges, by replacing the transverse spin by the longitudinal spin, e ⊥ · σ → e || · σ. Although we do not consider this process in this paper, a unique feature is that V and P Reggeon exchanges do not interfere in the forward amplitude due to the spin selection rule.
• So far, we have looked at V (= D * or K * ) meson production due to the planned experimental requirements. Theoretically, we can also study the reactions followed by D or K meson production. In this case, pseeudoscalar and scalar exchanges are possible, for which we can write down similar formulas.

IV. DISCUSSIONS AND REMARKS
We have studied charm production induced by the high-moment pion beam. This is a very challenging problem since no experiment has been performed for almost thirty years after the one at Brookhaven [9]. However, charmed baryon spectroscopy will bring us with fruitful information for yet unexplored region in hadron physics. This has been the primary motivation of the present study.
We have first estimated that in the Regge model charm production is suppressed by a factor 10 −4 as compared to strangeness production, implying an expected cross section of order 1 [nb]. Another yet important finding in the present study is that the production rates of excited charmed baryons are not necessarily suppressed as compared to those of the ground state. This is a consequence of good overlaps of the initial and final state baryons at the momentum transfer around 1 GeV, providing us with more opportunity for the study of excited states.
In the present study, we have used a simple quark and diquark model for baryons. In view of the successes of the constituent picture for low lying states, we expect some of the features should persist in the charm production reactions also. In particular, the identification of λ and ρ modes should be very important to reveal the mechanism of hadron excitations. Further investigations for productions and decays in the heavy quark region may provide good information of it.
for J = 1/2 and 3/2, and for J = 3/2. Here h denotes the helicity of the vector meson V . Other amplitudes are related to these elements by time reversal. The total cross section is then proportional to the sum of squared amplitudes over possible spin states. For J = 1/2 and for J = 3/2 and 5/2 First we consider the transition to Λ(1/2 + ) (of both charm and strangeness) where the baryon orbital wave functions ψ nlm are given in Appendix C. Note that since the diquark behaves as a spectator in the reaction (Fig. 1), the good diquark component of χ ρ for the nucleon is taken. The spectroscopic (overlap) factor of the good diquark component in the nucleon is tabulated in below where isospin factor is included also. Choosing the V polarization as e ⊥ , we have where the spin and orbital parts are separated and σ − is the spin lowering matrix given as The spin matrix elements are easily computed as where we have shown all relevant matrix elements in the following calculations. Therefore, the remaining is the elementary integral over the radial distance r with Gaussian functions. We find where the radial integral I 0 is given by The oscillator parameters are α and α ′ are for the initial and final state baryons, respectively. Similarly, we calculate the transitions to the ground state Σ's, picking up the χ λ part for the nucleon wave function. Only the difference is the spin matrix element which are computed by making Clebsh-Gordan decompositions. Results are where two independent matrix elements for Σ(3/2 + ) are shown.
Let us first consider the transition to Λ(1/2 − ). The rerelvant matrix element is given as where the factor 1/3 is the Clebsh-Gordan coefficients in the state [ψ 01 , χ ρ ] 1/2 −1/2 . The radial part is computed as and so Other matrix elements can be computed similarly: Computations go in completely similar manner as before, except for the radial matrix element ψ 020 | √ 2 e i q ef f · x |ψ 000 = 1 2 2 3 The results are [ψ 02 , χ ρ ] [ψ 02 , χ ρ ] [ψ 02 , χ λ ] [ψ 02 , χ λ ] We summarize the baryon wave functions used in the present calculations [23]. They are constructed by a quark and a diquark, and are expressed as products of isospin, spin and orbital wave functions. Here we show explicitly spin and orbital parts. For orbital wave functions, we employ harmonic oscillator functions as given in appendix C.
For spin wave functions, using the notation for angular momentum coupling [L 1 , L 2 ] Ltot we employ the three functions Similarly, we obtain the wave functions for the l = 2 excited baryons. Finally, the nucleon wave function is given as where φ ρ and φ λ are the ispsoin 1/2 wave functions of the nucleon with three quarks.