The nuclear matter effects in pi0 photoproduction at high energies

Abstract

The in-medium influence on pi0 photoproduction from spin zero nuclei is carefully studied in the GeV range using a straightforward Monte Carlo analysis. The calculation takes into account the relativistic nuclear recoil for coherent mechanisms (electromagnetic and nuclear amplitudes) plus a time dependent multi-collisional intranuclear cascade approach (MCMC) to describe the transport properties of mesons produced in the surroundings of the nucleon. A detailed analysis of the meson energy spectra for the photoproduction on 12C at 5.5 GeV indicates that both the Coulomb and nuclear coherent events are associated with a small energy transfer to the nucleus ($\lesssim$ 5 MeV), while the contribution of the nuclear incoherent mechanism is vanishing small within this kinematical range. The angular distributions are dominated by the Primakoff peak at extreme forward angles, with the nuclear incoherent process being the most important contribution above $\theta_{\pi^{0}}\gtrsim2^{0}$. Such consistent Monte Carlo approach provides a suitable method to clean up nuclear backgrounds in some recent high precision experiments, such as the PrimEx experiment at the Jefferson Laboratory Facility.

Meson photoproduction in-medium effects intranuclear cascade

The nuclear matter effects in p⁰ photoproduction at high energies

T. E. Rodrigues^I; J. D. T. Arruda-Neto^I; J. Mesa^II; C. Garcia^I; K. Shtejer^{I, III}; D. Dale^IV; I. Nakagawa^V

^IPhysics Institute, University of São Paulo, P.O. Box 66318, CEP 05315-970, São Paulo, Brazil

^IIDepartment of Physics and Biophysics - UNESP, Botucatu, SP, Brazil

^IIICenter of Applied Studies for Nuclear Developments (CEADEN), Havana, Cuba

^IVDepartment of Physics, Idaho State University, Pocatello ID, 83209, USA and 5RIKEN 2-1 Hirosawa, Wako, Saitama 351-0198, Japan

ABSTRACT

The in-medium influence on p⁰ photoproduction from spin zero nuclei is carefully studied in the GeV range using a straightforward Monte Carlo analysis. The calculation takes into account the relativistic nuclear recoil for coherent mechanisms (electromagnetic and nuclear amplitudes) plus a time dependent multi-collisional intranuclear cascade approach (MCMC) to describe the transport properties of mesons produced in the surroundings of the nucleon. A detailed analysis of the meson energy spectra for the photoproduction on ¹²C at 5.5 GeV indicates that both the Coulomb and nuclear coherent events are associated with a small energy transfer to the nucleus ($\lesssim$ 5 MeV), while the contribution of the nuclear incoherent mechanism is vanishing small within this kinematical range. The angular distributions are dominated by the Primakoff peak at extreme forward angles, with the nuclear incoherent process being the most important contribution above $\theta_{\pi^{0}}\gtrsim2^{0}$. Such consistent Monte Carlo approach provides a suitable method to clean up nuclear backgrounds in some recent high precision experiments, such as the PrimEx experiment at the Jefferson Laboratory Facility.

Keywords: Meson photoproduction in-medium effects intranuclear cascade

I. INTRODUCTION

The total p⁰ photoproduction cross section from complex nuclei can be evaluated as a sum of the electromagnetic part, known as the Primakoff effect[1], the nuclear coherent, the interference between the nuclear coherent and Primakoff amplitude and the nuclear incoherent contribution. The interference mechanism was neglected in this work since it consists of a genuine quantum process which is out of the context of the present semi-classical cascade calculation. While the electromagnetic part is almost non-sensitive to the Final State Interactions of the produced pions, the nuclear amplitudes are highly distorted by the medium and the complete nuclear ensemble should be taken into account in a realistic calculation.

In this paper we describe a sophisticated Monte Carlo method to address the dynamical evolution of an excited nuclear system. The calculation takes advantage of the Impulse Approximation (IA) to incorporate the interaction of the incoming photon with the nucleus in three different ways:

(1) a coherent electromagnetic interaction, where the pion is produced in the nuclear Coulomb field and, consequently, far away from the range of the nuclear potential which results in low Final State Interaction effects;

(2) a coherent nuclear production mechanism, where the pion is produced in the nuclear field and is susceptible to strong Final State Interactions, and;

(3) an incoherent photoproduction mechanism, where the pion is produced nearby a single nucleon, leaving the nucleus in an excited state and being also very sensitive to strong Final State Interactions.

These three scenarios apply, respectively, to the Primakoff, Nuclear Coherent and Nuclear Incoherent contributions for the total p⁰ production cross section and are described in more detail below.

The main focus of this work consists in the determination of the nuclear background in high energy-forward angles p⁰ photoproduction from spin zero nuclei, such as ¹²C and ²⁰⁸Pb. The calculation takes into account multiple scattering processes between the produced pions and the bound nucleons during the cascade stage. The Primakoff angular distribution is forward peaked because the Coulomb amplitude (g^* exchange) goes as 1/t, where t is the square of the four momentum transfer in the s channel (photoproduction channel) and also scales with the target charge and the square root of the p⁰® gg decay width. The measurement of this decay width, related to the gg^*p⁰ coupling constant of the Quantum Chromodynamics (QCD) has been the subject of various experiments, like the one performed at Cornell[2], and is also a major issue in a recent experiment carried out at the Jefferson Laboratory Facility, the PrimEx Collaboration[3]. In this regard, the measurements of the p⁰ angular distributions at forward angles, one of the PrimEx Collaboration's priorities, provide an indirect method to determine such an important and fundamental quantity.

II. THE INTRANUCLEAR CASCADE MONTE CARLO METHOD

The calculation outlined on this paper follows some previous work based on the Monte Carlo Multi-Collisional Intranuclear Cascade Model MCMC. Some improvements were incorporated to accommodate the Final State Interactions of the produced pions also for the nuclear coherent mechanisms, as well as the relativistic nuclear recoil for both the nuclear coherent and the electromagnetic (Primakoff) processes. The MCMC model was first developed to study high energy photonuclear reactions [4-7], being more recently successfully extended to intermediate energies[8, 9]. A special and very sophisticated version of the MCMC code were also proposed in a recent paper[10] in order to study the incoherent p⁰ photoproduction both at the D(1232) resonance energy region, as well as in the 4.0-6.0 GeV range and forward angles[10]. Such work showed the accuracy of the cascade model in predicting recent and intriguing results of the p⁰ differential cross section for ¹²C in the Delta region[11]. For details, see ref.[10] and references therein.

The MCMC model is essentially a typical transport calculation with some advantages related with the continuous time dependent structure available in the Monte Carlo analysis. Such an approach does not require a pre-determined time span between two successive collisions, like the well known BUU (Boltzmann-Uehling-Uhlenbeck) transport model [12, 13]. This special feature of the MCMC is related with the constant updating of the relativistic space-time, energy-momentum particle's four vectors along time.

III. THE p⁰ PHOTOPRODUCTION MECHANISMS

A. The electromagnetic p⁰ photoproduction mechanism

The differential cross section of the electromagnetic photoproduction of a pseudo-scalar meson P from a spin-zero nuclei may be written as[14]:

where G_P®gg, m_P, b and q_P are, respectively, the radiative decay width, mass, velocity and polar angle of the pseudo-scalar meson P. Z is the charge of the nucleus, F_el(Q) the nuclear electromagnetic form factor, k the incident photon energy and Q the four momentum transfer to the nucleus, with a = .

Eq.(1) indicates that the Coulomb production is forward peaked with its maximum at q_peak ~ . Such kinematical characteristic makes is possible to disentangle different production mechanisms in nuclei, shedding light to determine G_P®gg via angular distribution measurements.

Since the main subject on this paper is the nuclear contribution of the total p⁰ photoproduction, we have neglected the FSI effects in the Coulomb contribution wherever it appears. The relativistic nuclear recoil was taken into account applying the relativistic energy and momentum conservation to the photon - nucleus system, with the assumption that the whole nucleus has absorbed the momentum transfer Q, as it does in a coherent mechanism. With this approach, the complete kinematics of the interaction were determined by the meson polar angle, which was sampled in the Monte Carlo routine in accordance with eq.(1).

B. The nuclear coherent p⁰ photoproduction mechanism

Following the steps elegantly delineated in ref.[14], we may write the nuclear coherent cross section as:

where L is an angular-independent constant, A the nuclear mass and F_N(Q) the strong nuclear form factor. The constant L can be interpreted as a free parameter and was taken as L » 12.2k following the parameterization achieved for the Cornell data[2]. The nuclear coherent cross section peaks at approximately q_peak ~ , where R is the nuclear radius.

After being produced in the nuclear field, the pions are highly susceptible to FSI effects and may re-scatter in the nuclear matter, bringing the nucleus to an excited state. This process significantly changes the pion energies and angular distributions, attenuating eq.(2). Such a complex mechanism is out of the range of the conventional scattering theories, since various diagrams are relevant with the additional difficulty of dealing with many coupled channels. For this reason, the Monte Carlo method described here provides a powerful technique to address the correlation between coherent and incoherent processes via FSI.

The distortion effects on eq.(2) depend significantly on the nuclear volume and pion production energy. Besides it, nucleon-nucleon correlations due to the Pauli principle and the momentum distribution of the nucleons at the nuclear ground state play a key role for pion-nucleon scatterings at low polar angles (low momentum transfer).

C. The nuclear incoherent p⁰ photoproduction mechanism

The total incoherent production amplitude is characterized by a sum of single nucleon amplitudes. The photon probes a high density nuclear environment and is supposed to interact with a single nucleon, bringing the nucleus to an excited state. The fundamental property of such mechanism is that no interference occurs between the accessible states and the nuclear cross section simply scales with the nuclear mass. So, the total incoherent cross section may be written as:

where x is the p⁰ nuclear absorption, f is the Pauli-blocking suppression factor for small Q, is the photoproduction from the nucleon and l is a factor accounting for the p⁰ secondary production and re-scattering. The elementary photoproduction cross section can be written in terms of the helicity amplitudes, which are calculated in the context of the Regge theory under the assumption that the production process is largely dominated by vector meson (r and w) exchange for low t. Such elementary operator was calculated in ref.[10], reproducing quite succesfully the proton data at few GeV[15, 16]. For details on the calculations of the helicity amplitudes see ref.[10] and references therein.

The nucleon-nucleon correlations and the momentum distribution of the ground state determine the Pauli-blocking factor f. Indeed, this suppression function significantly changes eq.(3) for low momentum transfer and is the crucial step in reproducing the structures on the p⁰ differential cross section for ¹²C near the Delta resonance[10, 11].

IV. THE TRANSPORT OF PIONS THROUGH THE NUCLEAR MATTER

To access the Final State Interactions between the produced pions on their way out of the nucleus it is important to take into account the most relevant scattering processes for a given pN channel. This analysis was already performed in a previous work[10], where we have included the following pN channels for E_p< 6.0 GeV: i) elastic scattering; ii) resonance production and decay; iii) charge exchange processes; iv) pion re-absorption by a nucleon pair, and v) multiple pion production processes (up to seven particles in the final state). Such multiple scattering scenario dictates how the initial p⁰ flow is distributed among the available channels. In order to illustrate the rigorous Monte Carlo method to describe multiple pion-nucleon scatterings, we show in Fig. 1 the time derivative of the average number of pion-nucleon scatterings án_intñ with and without the Pauli-blocking for a typical cascade at 6.0 GeV for ¹²C and ²⁰⁸Pb. The dependence of on the nuclear volume is evident, showing that secondary scatterings are not relevant in light nuclei. The result for ¹²C exhibits a peak at the earliest cascade stages falling rapidly to zero around 5 fm/c. This gives a nice estimate ( ~ 5 fm) of the nuclear dimensions for a pion traveling at the speed of light. For the ²⁰⁸Pb, however, we obtained a much broader distribution with a maximum shifted towards ~ 5 fm/c, indicating that secondary scatterings with multiple pion productions are likely to occur for heavier systems.

V. RESULTS

A. p⁰ angular distributions for ¹²C at the GeV range and forward angles

With the prescriptions outlined above, we can calculate the angular distributions of neutral pions in the GeV range for the Primakoff, Nuclear Coherent and Nuclear Incoherent mechanisms. The results for ¹²C at k = 5.5 GeV in the laboratory frame are shown in Figs. 2, 3 and 4. Kinematical cuts on the total p⁰ energy were applied in order to estimate the nuclear background at different kinematical domains. The quantity DE represents the energy difference between the incoming photon and the outgoing pion: DE = k - E_p0. While for the coherent mechanisms (electromagnetic/nuclear) most of the pions have the total energy very close to the incident photon energy (DE 3 MeV), for the incoherent process (figure 4) there is a significant lowering on the pion energies caused by the requirement of a higher momentum transfer to excite the nucleus.

The Primakoff cross section (Fig. 2) exhibits a peak around ~ 0.017 degrees and falls rapidly to zero because of the dependence. It is interesting to note that almost all the pions have total energy between k and k - 1 MeV. This result reflects the forward peak constraint imposed by eq.(1).

The nuclear coherent cross section (Fig. 3) peaks at approximately [2/ kR] ~ 1.49 degrees, falling to zero for q_p⁰ >~3.5 degrees. Most of the pions produced via the nuclear coherent mechanism have total energy between k and k-3 MeV, in accordance with the kinematical characteristic inherent to eq.(2). The solid line of figure 3 represents the Plane Wave Impulse Approximation (PWIA), while the dashed line also includes the FSI effects of the produced pions.

The nuclear incoherent cross section is shown in Fig. 4. The PWIA is the solid line and the dashed line represents the calculations including the Pauli exclusion principle. The full calculation is the dotted line and takes into account the FSI effects. The structures of the cross section for low polar angles are attributed to the kinematics of the photoproduction channel, which takes into account the Pauli principle and a realistic momentum distribution of the nucleons at the ground state based on the shell model. For details on the momentum distributions see ref.[10]. The kinematical cuts on the incoherent cross section are very effective, showing that only few events can survive after a typical cut DE < 5 MeV.

B. p⁰ energy spectra for ¹²C at the GeV range and forward angles

The neutral pion energy spectra can be combined with the angular distributions to disentangle the different production processes in nuclei. The structure of the MCMC model permits a continuous evaluation of particle's energies along time, propitiating an accurate method to access information of an excited nuclear system.

Figure 5 shows the normalized p⁰ energy spectra for ¹²C at k = 5.5 GeV for the three different production channels. The Primakoff contribution (dotted line) exhibits a remarkable peak at DE < 1 MeV, as suggested by the angular distribution results. The nuclear coherent processes (dashed line) is also concentrated at the higher energy region, typically DE < 5 MeV, and dominates the photoproduction mechanism due to the small fraction of Primakoff events on the total (integrated) cross section. Both of the coherent processes are not relevant above DE ~ 6 MeV, since the respective form factors vanish at this kinematical domain ("high" momentum transfer region). The nuclear incoherent p⁰ energy spectrum (dashed-dotted-dotted line) is spread almost uniformly for lower pion energies. This contribution vanishes below the Primakoff peak as it demands a much higher momentum transfer in order to excite the nucleus. It is worth noting, however, that both the angular distributions and the pion spectra do not uniquely determine the type of photoproduction mechanism as different processes can take place at the same angular/energy interval. In fact, the nuclear incoherent contribution for high energy pions (DE 5 MeV) is substantially attenuated (figure 4) and the investigation of the pion spectra at these energies may serve as an additional constraint to clean up the incoherent contribution.

VI. CONCLUSIONS

A Monte Carlo method has been applied to study both coherent (Primakoff/nuclear) and incoherent p⁰ photoproduction mechanisms at the GeV range and forward angles from spin zero nuclei. The Coulomb process was analyzed taking into account the exact relativistic recoil of the nucleus and neglecting FSI effects of the produced pions (PWIA). The other processes, namely, the nuclear coherent and incoherent contributions were analyzed in the context of the MCMC intranuclear cascade model including full FSI effects.

The differential cross sections for ¹²C at 5.5 GeV are largely dominated by the Primakoff peak up to q_p0 0.5⁰, with the nuclear coherent mechanism being the most relevant contribution in the range 0.5⁰q_p0 2.0⁰. The nuclear incoherent part is Pauli-suppressed at lower angles and becomes the largest contribution for higher angles q_p0 2.0⁰. The energy spectra of the pions indicate that both the Coulomb and nuclear coherent processes are concentrated within a narrow energy range (DE 5 MeV) in contrast to the nuclear incoherent part, which is greatly inhibited in this energy gap due to a much larger momentum transfer to the nucleus.

In conclusion, a careful kinematical analysis using the MCMC model was performed to establish an efficient method of background subtraction for future reference in high energy p⁰ photoproduction experiments, such as the PrimEx Collaboration at the Jefferson Laboratory.

The authors thank the Brazilian agencies FAPESP and CNPq, and the Latin-American Physics Center (CLAF) for partial support to this work.

[3] http://www.jlab.org/primex/primex_notes/PR99-014.ps (PrimEx Proposal, accessed in Aug-2006)

[7] S. B. Duarte (private communication).

Received on 18 March, 2006

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Publication Dates

• Publication in this collection
  13 Apr 2007
• Date of issue
  Dec 2006

History

• Received
  18 Mar 2006