Mark Huyse
email: mark.huyse@fys.kuleuven.be

Riccardo Raabe
email: raabe@kuleuven.be

Radioactive Ion Beam

The first beam of post-accelerated radioactive ions was realized in 1989 at the Louvainla-Neuve research facility. The method employed two coupled cyclotrons to produce, separate and re-accelerate the species of interest. Several technological challenges were solved in the process, to obtain pure and intense beams for use in nuclear physics research. Similarly, new techniques were developed and refined for the measurement of the nuclear reactions induced by the radioactive beams. The available energy range made the facility particularly suited for nuclear astrophysics studies, and important results were obtained in the determination of stellar reaction rates using beams of 7 Be, 13 N, 18 F, 18,19 Ne. A beam of 6 He ions was extensively used in studies of the nuclear structure (halos, molecular states) and dynamics (the reaction process at energies around the potential barrier).

Introduction

Started in 1987 through a collaboration between the Belgian universities of Louvain-la-Neuve (UCL), Brussels (ULB) and Leuven (K.U.Leuven), the so-called RIB (Radioactive Ion Beam) project aimed to couple the two cyclotrons at Louvain-la-Neuve in order to produce an intense, energetic radioactive ion beam. On June 22, 1989, after two years of developments, a first beam was post-accelerated: a million 13 N ions per second were accelerated to an energy of 8 MeV [1]. In December 1990, a first experiment was carried out: in 33 hours of beam time the cross section of the proton-capture reaction on 13 N was measured using a 3 × 10 8 particles per s (pps), pure (∼99 %) 8.2 MeV 13 N beam [2]. Since then various beams of radioactive isotopes ranging from 6 He (T 1/2 = 0.8 s) to 35 Ar (T 1/2 = 1.7 s), with intensities from 1 × 10 5 pps ( 35 Ar) up to 2 × 10 9 pps ( 19 Ne), were developed and many experiments in nuclear astrophysics and nuclear structure were performed. Later on, the CYCLONE 44 cyclotron was designed and built specifically for experiments in nuclear astrophysics [3]. The RIB project came to an end in 2009.

Reaction studies with post-accelerated radioactive ion beams first of all require intense and pure beams at the right energy. The processes involved in arriving at this goal are the primary production in the target, the release out of the target, the transport to the ion source, the ionization to the desired charge state, the post acceleration and final beam manipulation. At the secondary target an efficient and selective detection set up which is able to cope with the intrinsic radioactivity of the secondary beam is needed. As the RIB facility at Louvain-la-Neuve was the first to deal with these issues, a description of the different developments undertaken are given in section 2. The following sections deal with nuclear astrophysics, nuclear structure and with reaction mechanisms around the Coulomb barrier studied at Louvan-la-Neuve using RIBs.

The production of RIBs at Louvain-la-Neuve

The layout of the RIB facility in Louvain-la-Neuve, which displays all the features of subsequent reaccelerated beam ISOL (Isotope-Separation-On-Line) facilities, is shown in Fig. 1. The coupled-cyclotron method is used to produce and post-accelerate the radioactive ion beams: the first cyclotron, CYCLONE 30, which is a 30 MeV proton accelerator, developed for medical purposes [4], produces the desired unstable isotope in suitable targets. Beam intensities up to 300 µA are used to irradiate the production target.

The resulting activity diffuses out of the hot target and is fed into an ECR ion source for ionization. The source operates at 6.4 GHz and is designed to produce with a high efficiency ions in relatively low-charge states [5]. After a first magnetic separation in a low mass-resolving dipole, the ions are injected in the second cyclotron, CYCLONE 110, for subsequent acceleration up to the desired energy. The publication, presenting the first beam obtained [1], describes all the ingredients of this process.

The primary production

The possibilities of producing radioactive species with the protons from CYCLONE 30 are limited to nuclei close to stability owing to the limited energy but one can produce very large quantities as the primary beam intensity can reach 300 µA. The choice of the production target is governed by the yield of the reaction, by the behavior of the target material under intense proton bombardment and by its release properties. The first beam developed at Louvain-la-Neuve was 10-min 13 N and its development will serve as a guideline to describe the whole RIB production process. A dedicated yield study considering two possible reactions for the production of 13 N was set up [6]. The chosen reaction (30-MeV p + 13 C) has a yield of 1.6 13 N nuclei per 1000 incoming protons.

Target-ion source system and development of some beams

Fast and efficient target-ion-source systems have been developed over the years at different ISOL facilities. However a special challenge was encountered at Louvain-la-Neuve as the power delivered in the thick target by the 300 µA 30 MeV proton beam reaches 6 kW, a record at that time. A robust target made of 13 C pellets, obtained by graphitizing 99% enriched 13 C powder was developed [7]. The release of the radioactive products out of the target was studied in off-line and on-line conditions [8]. The irradiated target not only releases the wanted radioactivity but also large quantities of unwanted radioactive and stable products. Chemical and/or cryogenic traps are used to reduce the contamination. The ion source needs to cope successfully with these hostile conditions in order to maintain its fast and efficient ionization. An electron cyclotron resonance (ECR) ion source was specifically developed for this and the efficiency and lingering time of different gasses studied [9]. In order to reduce the sticking, a quartz tube was installed in the source since quartz forms a rather inert surface. This resulted in an on-line efficiency for 13 N of 8% [9]. Another strong point of ECR sources is the relative high-charge state that can be reached with good efficiency. This facilitates the post-acceleration process considerably. Furthermore their ion-optical characteristics are well adapted for injection into cyclotrons. Next to the fully on-line approach where the production with the CYCLONE 30, the extraction out of the target, the ionization, mass separation and post acceleration with CYCLONE 110 all follows in a fast and continuous mode leading to beams of radioisotopes as short-living as 800 ms, a semi on-line mode was also developed for the production of 109.8-min 18 F [10] and an off-line batch method for the production of 53-days 7 Be [11]. The beams that were developed and post-accelerated at Louvain-la-Neuve are 6 He, 7 Be, 10,11 C, 13 N, 15 O, 18 F, 18,19 Ne and 35 Ar.

Purity of post-accelerated RIBs

Post-accelerated radioactive ion beams are mainly used for reaction studies. This imposes specific problems not encountered so dramatically with stable beams. RIBs are typically weak in intensity, sometimes contaminated and, of course, radioactive. The intensity depends on the reaction used and on the target-ion source system, both discussed above, and on the efficiency of the post acceleration. CYCLONE 110 reaches 10% efficiency for the acceleration of stable nitrogen while in on-line conditions the efficiency for 13 N is 4% [1]. The 13 N beam is 10 3 -10 4 times weaker than the isobaric contaminants 13 C and 12 CH. The relative mass differences with the 13 N ions are about 1.7 × 10 4 . The 12 CH 1+ molecular ions can be removed by employing a thin carbon foil where the molecular ions are dissociated and the products separated magnetically. The cyclotron can now be tuned as a radiofrequency mass spectrometer by reducing the acceleration voltage in order to increase the number of turns in the machine and therefore separates the 13 C contaminant from the 13 N beam. This method results in an isobaric suppression factor of 10 -6 while maintaining the efficient transmission through the cyclotron [12].

As the 13 N beam was the first post-accelerated ISOL beam developed it is instructive to examine Table I of the article describing the production process [1]. There a comparison is made between the off-line efficiencies of all the production steps and the final on-line efficiencies. Compromises have to be made when the different steps are brought together and a stable system running over several days is the final goal. The expected production by combining all off-line determined parameters was 6 × 10 10 pps while on-line 1.5 × 10 8 pps was routine. Such a reduction is normal and should be taken in account when planning new facilities or experiments. Finally these results could only be achieved through a close collaboration between the accelerator engineers and the physicists.

Nuclear astrophysics

Understanding the role of nuclear processes in astrophysical sites is a topic of great current interest and addresses questions such as how energy is generated in stars and how the chemical elements are synthesized. The advent of radioactive ion beams has created new opportunities to measure directly important capture reaction rates or to obtain crucial information through indirect methods. A recent review on the impact of experiments with radioactive ion beams on the field of nuclear astrophysics can be found in Ref. [13]. In this section we will concentrate on the methods that were specifically developed for the reaction studies with RIBs, on the pioneering experiments at Louvainla-Neuve related to the hot hydrogen burning CNO cycle and its link to the NeNa mass region, and will conclude with a short overview of recent experiments.

Methods

The reactions of astrophysical interest using RIBs in inverse kinematics impose specific requirements: firstly on the secondary target, and secondly on the detector set-up, which should be highly efficient as the secondary beam is in general of very low intensity and the reaction yield in the astrophysical energy domain is generally low. The detection set-up, however, must also cope with the radioactivity of the secondary beam. Finally reactions of astrophysical interest are typically measured in a well-defined energy domain.

As hydrogen and helium are the main ingredients of stars, reactions induced by these particles are of prime importance. However in order to study these reactions with RIBs, inverse kinematics has to be used implying the necessity of developing hydrogen and helium targets adapted to such experiments. For the 13 N(p,γ) 14 O reaction a feasibility study using the 13 C(p,γ) 14 N reaction was conducted comparing the resonance parameters obtained in inverse kinematics with an extended gas target (thintarget yield) and with a polyethylene foil (thick-target yield), with results from normal kinematics where thin and thick 13 C targets were bombarded with protons [14]. The loss in hydrogen of the polyethylene target was determined as a function of irradiation time and beam intensity (0.5% per hour per particle nA).

Helium targets were developed by implantation in thin Al foils, where concentrations of 10 17 -10 18 at/cm 2 [15,16] were possible, and by using a gas-filled chamber with thin windows [17]. The latter was tested with the 13 N(α,p) 16 O reaction and compared to the inverse reaction 16 O(p,α) 13 N.

There is no universal detection set-up for measuring reaction rates of astrophysical interest as all set-ups have to be tailored to the reaction under study. In the case of the 13 N(p,γ) 14 O reaction a large volume Ge detector was preferred to NaI(Tl) detectors to record the 5.173 MeV capture γ rays [9,12,14].

The amount of 20 Na produced in the 19 Ne(p,γ) 20 Na reaction was measured through the β + -delayed α radioactivity of 20 Na using two different methods and through the β + rays of 20 Na. The key problem was to detect the rare 20 Na radioactivity in the very high background arising from the decay of the unreacted 19 Ne beam. The first set-up for the β + -delayed α registration consisted of the on-line α detection in doublesided silicon strip detectors while the second set-up employed the off-line counting of nuclear tracks in polycarbonate solid state track detectors [18,19]. The detection of the β + decay of 20 Na was performed with the STAR set-up (Solenoid and Telescope for Astrophysical Research) where the combination of a solenoidal field and a stack of plastic scintillators were used to discriminate the β + signal from 20 Na in the background of the 19 Ne beam [20]. All set-ups were first tested with the 19 Ne(d,n) 20 Na reaction which has a considerably higher yield.

Another method which was also based on the detection of the radioactive end product of the reaction of interest (here the 13 N(d,n) 14 O reaction) consisted of detecting with well-shielded large-volume Ge detectors, the γ rays emitted in the β + decay of 14 O, implanted on a movable tape [21].

In a collaborative effort between the universities of Edinburgh and Louvain-la-Neuve, the very successful highly-segmented LEDA charged-particle array was developed to detect, in a strong β + background, relatively low-intensity charged reaction residues [22]. This approach has now been implemented at many places and has become one of the key experimental tools at such RIB facilities.

In a number of cases the sensitivity and selectivity of the detection set-ups described above were not sufficient and the Astrophysics REcoil Separator (ARES) was developed to measure (p,γ) reactions [23]. ARES has been coupled to CYCLONE44, the cyclotron purpose built for RIBs in the astrophysics domain [3].

In principle, in order to measure the proton resonance parameters when using a thin secondary target, the energy of the radioactive beam should be varied. This is a time-consuming process and a method was developed to scan in one measurement many resonances by using thick polyethylene targets [24,25]. The radioactive beam loses energy traversing the target: the initial beam energy and target thickness are chosen in order to cover the centre-of-mass energies of interest. The method, widely applied in Louvain-la-Neuve and other RIB facilities [2,25,26,27,28], has been extended to study the ground and excited states of unbound nuclei, either directly or through their isobaric analogue states: for example 11 N [29] and 19 Na [30,31] in Louvain-la-Neuve, 7 He [32] at the TwinSol Facility of the Notre Dame University. In addition, using α particles as a target, the same method was applied to the investigation of the cluster structure of resonances in selected nuclei (section 4.2).

First measurements

At the start of the RIB project, the primary focus was to measure cross-sections of astrophysical importance with special emphasis on capture reaction sequences from the hot CNO cycle towards the rp process. The reactions studied in Louvain-la-Neuve are shown in Fig. 2. The first result and also the first capture measurement with a radioactive ion beam was the direct determination of the 13 N(p,γ) 14 O reaction cross section which is crucial to determine the stellar conditions under which the hot CNO cycle competes with the cold CNO cycle [2,12]. In the relevant range of stellar temperatures the cross-section is dominated by the p = 0 resonance at E c.m. = 0.545 MeV corresponding to the 5.173 MeV (1 -) state in 14 O. The Γ γ width obtained resulted in a rate twice that previously adopted. In order to fully address the consequences one must also determine the interference of this resonance with the non-resonant capture. Therefore, the proton spectroscopic factors for the non-resonant contribution were measured with the 13 N(d,n) 14 O reaction [21]. No other experiments on the proton capture of 13 N were performed with post-accelerated ISOL beams elsewhere. This is probably due to the fact that the intensities reached at Louvain-la-Neuve (up to 15 particles pA) were already very high. This crucial reaction has been the object of further studies using the in-flight technique at Texas A&M [33], RIKEN [34] and Beijing [35], in particular on the non-resonant part. The dominant uncertainty in the 13 N(p,γ) reaction rate presently arises from the γ width of the 5.173 MeV state in 14 O. Shortly after these first experiments with the 13 N beam, a second important reaction was tackled. In a collaboration with the University of Edinburgh the influence of the 19 Ne(p,γ) 20 Na reaction on breakout of the CNO cycles into the rp process was studied. The reaction rate is believed to be dominated by a resonance at E c.m. = 0.447 MeV corresponding to the 2.643 MeV state in 20 Na, the first level above threshold in the 19 Ne(p,γ) 20 Na reaction. The 2.643 MeV state in 20 Na was previously observed in nuclear reactions induced by stable beams, such as 20 Ne( 3 He,t) 20 Na and 20 Ne(p,n) 20 Na, but the different results are in disagreement which leads to uncertainty in the spin assignment. Depending on the spin assignment the resonance strength estimates vary by almost a factor of 10, but all are considerably lower than the strength of the 13 N(p,γ) 14 O reaction. The resonant scattering in a thick target method described in section 3.1 was first employed to investigate resonances between 650 and 1000 keV [25]. Following this, a series of experiments, all based on the detection of the decay of 20 Na, were performed in the 1990's where more and more stringent limits were set on the resonant strength of the low-lying resonances in 20 Na [18,36,37]. These measurements suffered from a low detection efficiency (≤ 1.5%) and the presence of a background from the 19 Ne(d,n) 20 Na reaction induced by the beam on the deuterium present in the (CH 2 ) n target. The ARES spectrometer was used to overcome these problems and an upper limit of ωγ ≤ 15.2 meV for the resonance strength of the state at 447 keV was obtained [38]. The astrophysical implications of these studies show that the breakout of the CNO cycle to the rp process will be determined by the preceding 15 O(α,γ) 19 Ne reaction.

Further measurements

Following on from the pioneering work with 13 Ne, many more beams have been developed of which 7 Be, 18 F, 18 Ne have also been used for astrophysical studies. Furthermore, owing to the versatility of the LEDA array in its different configurations, not only could the direct measurement of (p,α) and (α,p) reactions be made, but resonance properties from elastic and inelastic scattering experiments could also be obtained (see also section 4).

The 7 Be beam produced in batch mode has made it possible to measure the 7 Be + p elastic cross section in the c.m. energy region from 0.3 to 0.75 MeV. From an R-matrix analysis of the cross section data, the energy and the width of the 1 + resonance in 8 B (E x = 0.77 MeV) was deduced and the astrophysical consequences discussed [39].

The 18 F beam was used for a direct measurement of the 18 F(p,α) 15 O reaction. This reaction, which influences the destruction of 18 F, is important for nova γ-ray astronomy as during nova outbursts the decay of 18 F is the main source of γ-ray emission at and below 511 keV. A series of experiments focussed on reactions with E c.m. above 550 keV [40,41,42]. Evidence for a resonance at 330 keV was found; the strength was later measured at the HRBIF facility [27]. The latest work at Louvain-la-Neuve yielded cross sections down to E c.m. = 400 keV constraining interference effects between 3/2 + resonances in 19 Ne [43]. However, recently a new experiment was performed with the 19 Ne beam to identify proton emitting states in 19 Ne through the inelastic scattering reaction 19 Ne(p,p) 19 Ne * (p) 18 F. Several proton-emitting states were identified and a new broad 1/2 + resonance was found, the low energy tail of which contributes to a significant enhancement of the 18 F destruction rate at nova temperatures (see above). This reduces significantly the chance to observe γ-ray emission of 18 F from a nearby nova explosion with existing telescopes [44].

The 18 F beam was also used in the 18 F(d,p) 19 F * reaction feeding levels above the α separation energy leading to 15 N + α. The resulting spectroscopic factors were used to put new limits to the contribution of low-energy resonances to the rate of 18 F(p,α) 15 O [45].

The 18 Ne beam was used in a study of the 18 Ne(α,p) 21 Na reaction, which can, at high stellar temperatures, be a possible breakout path out of the CNO cycle. Two centerof-mass energy domains were covered: between 2.04 and 3.01 MeV [17] and between 1.7 and 2.9 MeV [46]. Upper and lower limits on the reaction rates have been calculated from the resonance parameters of newly identified states.

Also using the 18 Ne beam, an attempt was made at studying the states in 19 Ne relevant for the 15 O(α,γ) 19 Ne reaction rate, by populating them through the d( 18 Ne, 19 Ne)p reaction [47]. With the increase in beam intensities expected at the forthcoming facilities, this method will become competitive with stable beam studies.

Nuclear structure

The contribution of radioactive ion beams to nuclear structure research has been particularly significant in the region of light nuclei. Nuclei at and beyond the driplines have become accessible using reactions with exotic beams. The proximity of the driplines has led to the discovery of a number of features, appearing in nuclei at excitation energies close to and beyond the breakup threshold.

Research in this region at Louvain-la-Neuve has been supported by the availability, since 1994, of a beam of post-accelerated 6 He, with an intensity and purity which were not matched elsewhere for many years. The beam has been directly used for investigations of the halo in 6 He and its role in the reaction process, but also as a tool to access exotic states and resonances in light systems.

The halo of 6 He

The research on the two-neutron halo of 6 He in Louvain-la-Neuve focused especially on the effects induced on the reaction mechanism; such aspects are discussed further in section 5. Here we mention two experiments aiming at the clarification of the structure of the halo, in particular the correlations between the two halo neutrons.

Halos had been initially studied through breakup reactions at intermediate and high energies, looking at the momentum distribution of the fragments [48,49]. The relatively low-energy post-accelerated beam at Louvain-la-Neuve opened the possibility for the use of transfer reactions. Because of the characteristics of the two neutrons in the halo -a low binding energy and a narrow structure of the wavefunction in momentum spacetransfer reactions with small momentum transfer and vanishing Q-value are favoured. The maximum cross section is thus obtained at low incident energies, for transfer into excited states of a stable target.

Given the α + 2n structure 6 He, it was natural to look into two-neutron transfer reactions, as they could be related to the spatial configuration of the two halo neutrons: the so-called di-neutron and cigar configurations [50]. Transfer reactions were performed on 12 C [51], 1 H [52] and 4 He targets [53,54]. All experiments employed the LEDA array; the former two focused on the study of the reaction mechanism, while the latter also investigated the importance of the transfer in relation to the 6 He configuration. The use of 4 He as a target had several consequences. Because of the symmetry of the system, the elastic scattering and the transfer of two neutrons would contribute to the same exit dσ/dΩ (mb/sr) 4 He( 6He, 6 He) 4 He E c.m. = 11.6 MeV Figure 3. 4 He( 6 He, 6 He) 4 He cross section at E c.m. = 11.6 MeV, measured using a gas target (open dots) [53] and a 4 He-implanted Al target (full dots) [54]. The curve is a coherent sum of elastic scattering and transfer amplitudes, the latter obtained with a DWBA calculation.

channel (a 6 He nucleus and an α particle), with the transfer manifesting itself through interference phenomena and especially a rise of the elastic scattering cross section at backward centre-of-mass angles. A gas cell was first employed, but this limited the angular range at which the outgoing particles could be detected. This improved with the used of a 4 He-implanted Al foil [15], a novelty at the time in combination with a radioactive ion beam. The second measurement indeed revealed 6 He + α events at large centre-of-mass angles (see Fig. 3) proving the occurrence of two-neutron transfer.

In the original work [54], data were reasonably well reproduced by a distorted-wave Born approximation (DWBA) calculation, assuming a direct transfer of two neutrons as a cluster. Later works used these data and those obtained at a higher energy at the Flerov Institute in Dubna [55], to investigate in more detail the possible sequential transfer of the two neutrons, which could have originated from a cigar configuration of the halo neutrons. In both Refs. [56] and [57] it was concluded that the direct transfer is the dominant mechanism, allowing the weight of the di-neutron component in the halo to be estimated to be around 95% [57].

Another process which was identified as strongly dependent on the details of the halo is the β-decay of 6 He, in particular the deuteron-emission channel, first observed at ISOLDE [58]. The reason of the very weak branching ratio (∼10 -6 ) was explained in Ref. [59] in terms of a cancellation between two parts of the Gamow-Teller matrix element, related to the internal and external (halo) regions of the 6 He ground state and the α + d scattering wave functions. Measurements of the branching ratio performed at ISOLDE [58,60] and TRIUMF [61] using low-energy beams suffered from normalization uncertainties. In Louvain-la-Neuve, the pure post-accelerated beam permitted the technique of direct implantation in a silicon detector [62], where decays could be counted with 100% efficiency providing an accurate normalization, to be employed. A problem that had to be overcome was the instantaneous suppression of the beam to a level much better than the branching ratio, 10 -6 , required in order to observe the decay events without beam-induced background. This was achieved by a combination of collimators and a rotating shutter wheel. The branching ratio (1.65 ± 0.10) × 10 6 [63] was measured with an uncertainty that represented a significant improvement with respect to previous results. This, and the related deuteron energy spectrum obtained, can provide accurate tests of models of the 6 He ground-state wave function [63,64]. The techniques developed here were later employed at TRIUMF to study the β-delayed deuteron emission of 11 Li [65].

Cluster states

The possibility of states in light nuclei, with configurations reminiscent of molecular structures, was advanced as early as 1938 [66]. In such cases, nucleons exchanged between different α clusters ensure the binding of the system. The existence of these structures was shown in 9 Be through the identification of rotational bands built on the deformed molecular α:n:α state [67,68]. An analogue identification of states in 10 Be has been more elusive. Of particular interest would be a α:2n:α state in which the two neutrons occupy a σ 2 valence orbital: owing to Pauli blocking, the neutrons would push the α clusters apart resulting in a very deformed state. A 0 + state at E * = 6.18 MeV was the main candidate, with the 2 + excited state of the corresponding rotational band at E * = 7.54 MeV. But conclusive evidence could only come from the identification and characterization of the 4 + state.

This was obtained in two separate measurements at Louvain-la-Neuve. A first indication came from the study of the 6 He + 6 Li reaction at 18 MeV beam energy [69]: a state at 10.2 MeV in 10 Be, populated by the transfer of an α particle from the target and identified via the coincident detection of the outgoing 6 He, α and d particles, was assigned as 4 + by analysing the angular correlation between the products. A following experiment [70] concentrated on the characteristics of this resonance, which was studied using the method of resonant elastic scattering in a thick gas target. A 7.5-MeV 6 He 1+ beam impinged on a volume of 4 He gas; the beam particles lose energy traversing the gas volume, undergoing elastic scattering with enhanced probability for center-of-mass energies corresponding to the resonance (see Fig. 4). The coincidence detection of the 6 He and 4 He particles and their angular correlation allowed to definitely assign the 4 + spin to the state, identifying it as a member of the rotational band. The cross section indicated a very large α-particle component of the state (one of the largest α-cluster [8] were mounted within the reaction chamber (Fig. 1) which was filled with helium gas (150 mb). These arrays are composed of 16-element annular strip detectors with an inner radius of 50 mm and an outer radius of 130 mm. The The region in the gas volume in which the 10.15 MeV 10 Be resonance is formed is indicated (shaded zone). The resonance subsequently decays into 6 He 4 He which are detected in coincidence in the LAMP array.

501-1

© 2006 The American Physical Society (ÿ ) of the resonance. For this analysis the data in the range c:m: 70 and 110 , where the effects of the experimental acceptances are reduced, were used. From the simulations it was determined that the efficiency for detecting, in coincidence, the 6 He 4 He pairs is 1.37(0.07)%. This compares with 7.2% of the fraction of the P 2 4 Legendre polynomial which lies within the same quence of the 6 He and 4 He. good at small indicates that are not so we the distributio spectroscopic factors known to date). Finally, from its energy, measured at 10.15 MeV, a large moment of inertia was derived, confirming the very deformed α:2n:α nature of the state at 6.18 MeV. The same experimental technique was used with the 18 Ne beam to investigate αcluster states in 22 Mg [71]. A comparison with mirror states in 22 Ne showed significant, poorly understood differences.

In other measurements, these structures in light nuclei were investigated by identifying "quasi-free" scattering events on clusters of the target. This was for example the case for the scattering of 6 He off deuterons in 6 Li [72] and off α particles in 9 Be [73].

In-beam spectroscopy using RIBs

Thanks to the early availability of very intense post-accelerated radioactive ion beams, Louvain-la-Neuve has been a favourite place to test novel techniques in nuclear spectroscopy. An interesting example is provided by the γ-ray spectroscopy of nuclei produced in fusion-evaporation reactions. The high-spin states in such nuclei are usually reached using intense stable beams and different regions of the nuclear chart have been studied that way. However, some regions are not easily accessible using stable targetbeam combinations. As an example, the study region of the heavy, unstable N ∼Z nuclei is hindered by the very low cross sections for multiple neutron evaporation. The use of a proton-rich radioactive ion beam could be an advantage as the corresponding cross sections are higher, provided that this increase compensates the lower beam intensity. An additional problem of such an approach is represented by the high 511-keV background in the γ detectors, originating from the β + decay of the radioactive beam.

A measurement was performed using a very intense (10 9 pps) 70-MeV 19 Ne 4+ beam impinging on a 40 Ca target [74,75], with the purpose of investigating these issues and test possibilities for reducing the background. The γ rays produced in fusion-evaporation reactions were detected in an array of escape-suppressed Ge detectors surrounding the target. Additionally, the LEDA array was employed to detect evaporated light charged particles.

Several conclusions were drawn: it was necessary to carefully shield the Comptonsuppression detectors from the radiation produced by the decay of beam particles hitting collimators close to the target; from this, the importance of a very-well focused beam without a halo was identified. The charged-particle array proved very effective in identifying the reaction channels; here, particle discrimination (protons and α's) was crucial, determining an ideal thickness of the detector of about 150 µm. Finally, the most effective background suppression was obtained by the time correlation between the γ rays and the beam particles. To achieve this, the beam was pulsed in 2 ns-wide bursts separated by 73 ns. The γ-ray spectra measured in between pulses were scaled and subtracted from the coincidence spectra, eliminating most of the random background. The effect is clearly visible in Fig. 5. At present, this technique is widely used at RIB facilities, including, for example, REX-ISOLDE to reduce the background in the Miniball γ-ray array in Coulex and transfer measurements.

Reaction mechanism around the barrier

As already recognized with the measurements that led to the discovery of the halo [76,77], light exotic nuclei have properties that significantly influence the reaction mechanism. The characteristics of these nuclei could enhance the cross section for selected direct channels (for example, breakup or neutron transfer). At energies around the potential barrier, where the terms connecting the different exit channels of the scattering are no longer negligible with respect to the kinetic energy of the particles, coupling effects become important modifying the probability for elastic scattering [START_REF] Satchler | Introduction to nuclear reactions[END_REF] and fusion [START_REF] Beckerman | [END_REF]80]. The latter process in particular was the centre of much interest shortly after the discovery of halo nuclei, as it was speculated that the large spatial extension of neutron matter could lead to a large enhancement of the fusion probability.

The measurements related to these features are extremely challenging. For elastic scattering, a high accuracy is required, to allow to be compared with models which take into account the various coupling effects. For fusion, the small cross section around and below the potential barrier makes it inherently difficult to perform studies using the relatively low-intensity radioactive ion beams. For this reason, only limited regions of the excitation functions have been explored. Nevertheless, significant results have been achieved: recent reviews are given in Ref. [81] for elastic scattering and Refs. [82,83] for fusion. The facility at Louvain-la-Neuve has contributed significantly to measurements using the 6 He (see below) and 7 Be beams [84]. [74]). γ-ray energy spectra from the 19 Ne + 40 Ca reaction. Panel (d) shows the dramatic improvement when the random background is subtracted using the time correlation with the beam pulses, as compared to singles (a), coincidence with light particles (b) and γ-γ coincidences (c).

Elastic scattering

Elastic scattering of the 6 He beam on various thin targets were already obtained as a by-product of the first transfer measurements in the second half of the 1990's. These data have been the object of some re-analyses [51,85,86,87,88].

Those first measurements were carried out using the LEDA array, covering only forward angles up to about 75 degrees. Improved, dedicated measurements [89] employed additional detectors at backward angles to increase the angular coverage. The focus has been on the scattering of 6 He on 208 Pb: the target was chosen because of its relatively simple and well-known structure, and because it would enhance Coulomb effects. The [92]). Schematic view of the set-up used in the measurement of the 6 He + 238 U reaction. For clarity, some of the forty detectors are omitted in the drawing.

elastic cross section was measured at various centre-of-mass energies between 14 and 22 MeV and angles up to 170 degrees, and analyzed using optical model calculations. The data showed significant differences with respect to those obtained with 6 Li: in particular evidence for long range absorption was found, due to a polarization of the projectile in the Coulomb field of the target. A separate study of the outgoing α particles [90] showed the presence of a strong two-neutron transfer from the 6 He projectile to weakly bound states in 210 Pb.

The peculiarities in the elastic scattering of 6 He were also clearly highlighted in the comparison of 6 He + 64 Zn and 4 He + 64 Zn data, obtained in a campaign where the fusion cross section was also measured [91].

Fusion measurements

The measurement of the fusion cross section of the 6 He represents one of the highlights of the research at Louvain-la-Neuve using radioactive beams. Data were obtained on two targets, 238 U and 64 Zn, utilizing different techniques. In both cases, the simultaneous measurement of large cross sections due to direct channels showed their importance at energies even below the barrier.

In each case, the nature of the target provided the means to identify the fusion process. The low fission barrier of 238 U meant that reactions in the thin UF 4 target and the 6 He projectiles led to the back-to-back emission of two fission fragments. The fragments were identified by a coincident detection in a silicon array covering almost the full solid angle (Fig. 6). Fission could be induced by fusion, but also by other processes transferring sufficient energy to the target; in the latter cases, however, an additional light quasi-projectile particle would be emitted in coincidence and detected in the silicon array. This provided a way to separate the contribution of fusion from that of other direct processes. A first measurement with a reduced angular coverage showed a very large fission cross section [93]. The addition of detectors at backward angles proved, however, the strong presence of direct processes, reducing the fusion cross section to levels comparable with those expected for a well-bound, non-halo projectile such as the stable 4 He [92].

With the 64 Zn target nucleus, an activation technique, followed by the off-line detection of residues, was employed. The 64 Zn provided the possibility of uniquely identifying the reaction products. In this experiment a stack of Zn targets was used, alternated with Nb foils that would catch the nuclei recoiling from the target (Fig. 7). Owing to the energy loss of the 6 He projectiles in traversing the stack, the reactions on each target would provide the cross section at a different energy, allowing the excitation function to be explored with only a few primary beam energies. An additional thin target, placed before the stack and surrounded by silicon detectors, served for the measurement of the elastic scattering as mentioned in the previous section [91]. A fusion reaction within the stack would eventually produce radioactive, longliving Ge, Ga and Zn evaporation residues. The atomic activity was identified and counted through the detection of the x-ray emission following the electron-capture decay; further identification of the specific isotope was obtained by its characteristics half-life signature. The results, reported in Ref. [94], confirmed that which was observed with the heavier 238 U target: the data for the fusion cross section did not show any appreciable enhancement with respect to those measured with 4 He. On the other hand, a very large yield for 65 Zn, not reproduced by fusion-evaporation calculations, could be explained by a large transfer and/or breakup cross section.

Conclusions

With the first post-acceleration of a radioactive ion beam, 13 N, research in nuclear physics acquired a powerful new tool to explore regions of the chart of nuclei far from stability. Coupling of the ISOL method for the on-line production and selection of ions with an efficient post-accelerator provided beams of unprecedented intensity and optical properties. To reach this seminal result, a number of technologies and solutions were conceived and pioneered at the Louvain-la-Neuve facility. These included methods to cope with the high power dissipated by the primary proton beam in the thick production target; research, on target materials and on suitable reactions, to obtain high production rates for unstable isotopes; and the means for optimizing the beam quality, in particular its purity, without compromising the overall acceleration efficiency. This experience has especially shown how entangled all these aspects are, and thus how the development of each beam requires a dedicated and specialized effort. This is an effort that has to be undertaken in close collaboration between the accelerator engineers and the physicists as the beam quality determines the detection set-up to be developed and vice-versa.

The development of new beams was driven by the physics program. Owing to the energies provided by the post-accelerator, nuclear reactions of astrophysical interest occurring at stellar temperatures were accessible. Important measurements were made of rates of reactions in the hot-CNO cycle and escapes towards the rp process; clearly, such reactions -like 13 N(p,γ) 14 O, 19 Ne(p,γ) 20 Na -were measured directly for the first time. A major role was then played by the 6 He beam, for research in nuclear structure and dynamics. On one hand, the focus has been on the characteristics of the two-neutron halo of 6 He and its influence on the reaction process at energies around the Coulomb barrier. On the other hand, the beam was used as a tool in reactions to produce systems in exotic states and resonances, such as the molecular structure of the very deformed α:2n:α cluster state identified in 10 Be.

As well as the beam production techniques, detection methods were developed and applied to counter the problems presented by the use of radioactive beams. The method of resonant scattering in a thick target in inverse kinematics was developed using solid targets (polyethylene, He-implanted foils) as well as He gas targets. The issues related to in-beam γ-ray spectroscopy with RIBs were carefully studied, producing reference results relevant for the subsequent design of optimized detection set-ups and experiments elsewhere.

Several of the methods pioneered in Louvain-la-Neuve have been taken up and further improved at the more recently-developed RIB facilities. With the progress of the latter, the activities in Louvain-la-Neuve have lessened; this is true in particular for nuclear structure research, which at present requires an increasingly strong concentration of resources. The RIB project came to an end in 2009. The experience accumulated in more than twenty years of target and ion source development continues, however, to be exploited by participation in projects such as beta-beams [95]; in particular for the aspects related to the production of extremely intense 6 He and 18 Ne beams -for which Louvain-la-Neuve has long held the records, and also for the more recent schemes involving 8 Li and 8 B beams [96], where, once again, major technological challenges will need to be overcome.

Figure 1 .

 1 Figure 1. Layout of the radioactive ion beam facility in Louvain-la-Neuve. Both the CYCLONE 110 and CYCLONE 44 cyclotrons post-accelerated radioactive nuclei produced by reactions induced by the proton beam of CYCLONE 30.

Figure 2 .

 2 Figure 2. Portion of the chart of nuclei, with indication of the paths of the hot-CNO cycle (Hot-CNO I) and very-hot CNO cycle (Hot-CNO II), with the possible escape routes. The reactions studied in Louvain-la-Neuve are highlighted.

FIG. 1 .

 1 FIG.1. Experimental setup. The region in the gas volume in which the 10.15 MeV 10 Be resonance is formed is indicated (shaded zone). The resonance subsequently decays into6 He4 He which are detected in coincidence in the LAMP array.

FIG. 3 (FIGFigure 4 .

 34 FIG. 3 (color online). Experimentally determined center-ofmass distribution (data points) compared with simulations for the decay of a spin 4 (blue-solid line), 2 (red-dashed line), and 6 (black-long-dashed line) resonance. The inset shows the energyspin systematics of the rotational band of which the present 4 state is a member.

Figure 5 .

 5 Figure 5. (from Ref.[74]). γ-ray energy spectra from the19 Ne + 40 Ca reaction. Panel (d) shows the dramatic improvement when the random background is subtracted using the time correlation with the beam pulses, as compared to singles (a), coincidence with light particles (b) and γ-γ coincidences (c).

Figure 6 .

 6 Figure 6. (from Ref.[92]). Schematic view of the set-up used in the measurement of the6 He + 238 U reaction. For clarity, some of the forty detectors are omitted in the drawing.

Figure 7 .

 7 Figure 7. (from Ref.[94]). Schematic view of the set-up used in the measurement of6 He + 64 Zn. LAMP and DSSD are arrays of silicon detectors.

  14 O

		Hot-CNO I				24 Al	25 Al
		Hot-CNO II
		Breakout Breakout
		rp-process		21 Mg	22 Mg	23 Mg	24 Mg
		measured in
		Louvain-la-Neuve
				20 Na Na	21 Na Na	22 Na Na	23 Na Na
			18 Ne	19 Ne	20 Ne	21 Ne	22 Ne
			17 F	18 F	19 F
	14 O	15 O	16 O	17 O	18 O
	13 N	14 N	15 N
						stable
	12 C 12 C	13 C 13 C				t bl unstable

Acknowledgments

The authors wish to thank the staff of the Cyclotron Research Centre in Louvainla-Neuve for their continuous effort in developing and delivering excellent radioactive ion beams and for their warm hospitality in receiving the many outside users. We also wish to thank our colleagues from UCL, ULB and K.U.Leuven who formed the enthusiastic core of the facility. Finally, we acknowledge the continuous support from the Interuniversity Poles of Attraction Program from the Belgian State and from the various European programs supporting experiments at the facility.


N Kaiser

Radiative corrections to electron-pion scattering revisited

Keywords: 12.20.-m, 12.20.Ds, 14.70.Bh 1

We calculate in closed analytical form the one-loop radiative corrections to electronpion scattering e -π -→ e -π -. Concise expressions (in terms of dimensionless Mandelstam variables) are given for the pertinent interference terms between the tree diagram and the one-loop diagrams related to photonic vertex correction, vacuum polarization and two-photon exchange. Infrared finiteness of these virtual radiative corrections is achieved (in the standard way) by including soft photon radiation below an energy cutoff λ. We evaluate the finite part of the soft photon correction factor in the center-of-mass frame. For all contributions we keep the full dependence on the electron mass. The results for the loop amplitudes can be applied also to pion-pair production e -e + → π + π -. The radiative corrections to the corresponding total cross section are calculated with consideration of the pion-structure via a vector meson dominance form factor. We find that such an additional charge form factor (inserted into the photonic vertex correction of the pion) changes the radiative corrections to the total cross section by about -0.2%.

Introduction and summary

For a correct interpretation of the experimental data obtained in elastic or inelastic lepton scattering processes it is essential to include in the analysis of these data the radiative corrections arising from virtual photon-loops and (soft) photon bremsstrahlung. The subject of radiative corrections has by now already a long history and we refer to the standard review papers of Maximon [1] and Mo and Tsai [2] which have been used in the interpretation of many electron scattering experiments. In recent years the issue of radiative corrections has received a renewed interest in connection with new precision experiments, performed e.g. at JLAB or MAMI. The two-photon exchange contribution [3,4,5] to the elastic electron-proton scattering plays in fact a crucial role in order to reconcile the apparent discrepancies for the ratio of the proton electric and magnetic form factors G p E (Q 2 )/G p M (Q 2 ) as determined with the polarization transfer technique on the one hand side and via the (more traditional) Rosenbluth separation method on the other hand side.

These findings have prompted the exploration of the significance of two-photon exchange in other electromagnetic reactions (see ref. [6] for a recent review on this subject). In particular, the two-photon exchange corrections to elastic electron-pion scattering have been calculated in ref. [7] taking into account the finite size of the pion through a (phenomenological) monopole form factor F π (Q 2 ) = (1 + Q 2 /m 2 ρ ) -1 . It has been found, that in comparison to the softphoton approximation which neglects hadron structure effects, the corrections are less than 1% for low momentum transfers Q 2 < 0.1 GeV 2 , but can increase to several percent for Q 2 ≥ 1 GeV 2 at extreme backward angles. The analytical expression for the two-photon exchange amplitude with insertion of monopole form factors at the photon-pion vertices has however (A) Figure 1: One-photon exchange tree diagram (A) for electron-pion scattering. Full, dashed and wiggly lines denote electrons, pions and photons, respectively.

Radiative corrections to electron-pion scattering

We start out with calculating the radiative corrections to electron-pion scattering. The inand out-going four-momenta of the reaction e -(p 1 ) + π -(p 2 ) → e -(p 3 ) + π -(p 4 ) give rise to the Lorentz-invariant (dimensionless) Mandelstam variables (s, t, u) defined as follows:

s m 2 π = (p 1 + p 2 ) 2 , tm 2 π = (p 1 -p 3 ) 2 , um 2 π = (p 1 -p 4 ) 2 . ( 1
)
Since m 2 π has been factored out, they obey the numerical constraint s + t + u = 2 + 2r with r = (m e /m π ) 2 the squared ratio between the electron mass m e = 0.511 MeV and the pion mass m π = 139.57 MeV. In the case of unpolarized scattering the squared T-matrix for e -π -→ e -π -has to be summed over the electron spin-states. This double sum is efficiently performed via a Dirac-trace:

1 8 tr[O(/ p 1 + m e )O(/ p 3 + m e )].
In order to keep the technical complication of the loop calculation as low as possible the integration over the loop-momenta is carried out after the summation over the electron spin-states. With that approach in mind the unpolarized differential cross section for electron-pion scattering in the center-of-mass frame, including radiative corrections of relative order α, can be represented in the following compact form:

dσ dΩ cm = α 2 m 2 π s A ⊗ A t 2 + 2 t Re (I + II + III + IV + V) ⊗ A , (2)
with α = 1/137.036 the fine-structure constant. Here A denotes the one-photon exchange tree diagram (shown in Fig. 1) and I, II, III, IV, V stand for the five (classes of) contributing one-loop diagrams (shown in Figs. 2,3,4). The product symbol ⊗ designates the interference term between the T-matrices from two diagrams with the sums over the electron spins already carried out via a Dirac-trace. When written in terms of the dimensionless variables (s, t, u, r) the tree diagram A in Fig. 1 leads to the following simple polynomial expression:

A ⊗ A = 1 -(s -r)(u -r) . (3)
Note the invariance of A⊗A under the crossing transformation s ↔ u. It expresses the obvious fact that at leading order the differential cross sections for e -π -→ e -π -and e -π + → e -π + are equal.

Evaluation of one-loop diagrams

In this section, we present analytical expressions (of order α) for the interference terms between the tree diagram and the one-loop diagrams for electron-pion scattering. We use dimensional

(I e ) (I π ) regularization to treat both ultraviolet and infrared divergences (where the latter are caused by the masslessness of the photon). Divergent pieces of one-loop integrals show up in form of the composite constant:

ξ = 1 d -4 + 1 2 (γ E -ln 4π) + ln m e µ , (4)
containing a simple pole at d = 4 and µ is an arbitrary mass scale. Ultraviolet (UV) and infrared (IR) divergences are distinguished by the feature of whether the condition for convergence of the d-dimensional integral is d < 4 or d > 4. We discriminate them in the notation by putting appropriate subscripts, i.e. ξ UV and ξ IR . 1 In order to simplify all calculations, we employ the Feynman gauge where the photon propagator is directly proportional to the Minkowski metric tensor g µν . Let us now enumerate the analytical results as they emerge from the five (classes of) one-loop diagrams shown in Figs.

The (on-shell) vertex correction in diagram I e is comprised by the one-photon loop form factors F 1,2 (t) γ-loop of the electron. These Dirac and Pauli form factors are normalized at t = 0 to: F 1 (0) γ-loop = 0 and F 2 (0) γ-loop = α/2π (anomalous magnetic moment). Putting all the pieces together the pertinent interference term (A ⊗ A)F 1 (t) γ-loop /t + (1 -t/4)F 2 (t) γ-loop with the tree diagram A reads:

I e ⊗ A = α 2πt 4ξ IR (t -2r) + 8r -3t L(-tr -1 ) √ 4r 2 -rt + 2ξ IR -2 + (t -2r)Φ(-tr -1 ) √ -t √ 4r -t (A ⊗ A) + √ r(4t -t 2 ) L(-tr -1 ) √ 4r -t , (5)
with A ⊗ A given in eq.( 3). Here, we have introduced the frequently occurring logarithmic loop function:

L(x) = 1 √ x ln √ 4 + x + √ x 2 , (6)
and the auxiliary function:

Φ(x) = Li 2 (v(x)) -Li 2 (1 -v(x)) + 1 2 ln 2 v(x) - 1 2 ln 2 (1 -v(x)) , (7)
composed of dilogarithms and squared logarithms of the argument:

v(x) = 1 2 1 - x 4 + x , (8)
(II e,µ )

(II π ) where Li

2 (v) = ∞ n=1 n -2 v n = v ∞ 1 dy[y(y -v)] -
(-t) √ -t √ 4 -t = ∞ 4 dx x -t ln(x -4) √ x 2 -4x . ( 9
)
Note that in addition to the photon-loop also the counterterm Z (e)

1 -1 = α(2ξ IR + ξ UV -2)/2π of spinor quantum electrodynamics, which eliminates the ultraviolet divergence ξ UV in the Dirac form factor F 1 (t) γ-loop , has been included in eq.( 5).

In the same way, the (on-shell) vertex corrections in diagrams I π are comprised by the one-photon loop form factor G π (t) γ-loop of the pion. When completed by the counterterm contribution Z (π) 1 -1 = α(ξ IR -ξ UV )/π of scalar quantum electrodynamics this form factor is also normalized to G π (0) γ-loop = 0. The pertinent interference term (A ⊗ A)G π (t) γ-loop /t with the tree diagram reads:

I π ⊗ A = α 2πt 2ξ IR -2 -ln r + t -2 √ -t √ 4 -t Φ(-t) +(4ξ IR -4 -2 ln r) √ -t L(-t) (A ⊗ A) . (10)
The additional ln r terms are present because the infrared divergence ξ IR has been defined in eq.( 4) with the electron mass in the logarithm ln(m e /µ).

The diagrams of class II shown in Fig. 3 involve vacuum polarizations at the exchanged photon. Together with the counterterm Z (e) 3 -1 = 2α ξ UV /3π the (renormalized) contribution from electronic vacuum polarization takes the form:

II e ⊗ A = α 3πt 2 2 √ r (t + 2r) √ 4r -t L(-tr -1 ) - 5t 3 -4r (A ⊗ A) . (11)
The analogous contribution from muonic vacuum polarization II µ ⊗ A is readily obtained by replacing the parameter r in eq.( 11) by r = (m µ /m π ) 2 = 0.57309. In addition there is the pionic vacuum polarization diagram II π . Together with the tadpole diagram (generated by the γγππ contact-vertex) and the counterterm Z (π) 3

-1 = α(ξ UV -1 2 ln r)/6π the complete (renormalized) contribution from pionic vacuum polarization reads:

II π ⊗ A = α 6πt 2 4 - 4t 3 + (t -4) √ 4 -t L(-t) (A ⊗ A) . (12)
Next, we come to the two-photon exchange triangle diagram III shown in Fig. 4. The respective loop integral with one massive (pion) and two massless (photon) propagators is infrared convergent and the pertinent interference term with the tree diagram takes the form:

(III) (IV) (V)
III ⊗ A = α π s -u 4r -t (4r -2t)K(tr -1 ) + r ln r -ln(-t) , (13)
with the t-dependent auxiliary function:

K(t) = 1 √ -t √ 4 -t π 2 3 -t L 2 (-t) + Li 2 2 -t - √ -t √ 4 -t 2 . ( 14
)
Somewhat more intricate is the evaluation of the (planar and crossed) two-photon exchange box diagrams IV and V shown in Fig. 4. We take advantage of performing first the spinsums (via a Dirac-trace) and of decomposing in the next step the resulting Lorentz-scalar loop integrand into partial fractions. In this form the majority of terms has either only one electron propagator or only one pion propagator and few terms involve the product of both. The latter loop integrals depend in a non-trivial way on the squared mass ratio r = (m e /m π ) 2 . Putting all the pieces together one finds for the interference term of the planar two-photon exchange box diagram IV with the tree diagram the following analytical expression:

IV ⊗ A = α π s 2 (s -2r)K(t) + s 2r (s -2 -2r)K(tr -1 ) + 1 + r -s √ ρ + -s √ ρ --s (4ξ IR -2 ln r) s -r + 1 t (s -1 -r) 2 + s -r + 2 t (s -1 -r) 2 ln(-t) ln √ ρ + -s + √ ρ --s 2r 1/4 + 1 2 (s -r)(1 + r -s) ∞ ρ + dx x -s ln[x -2 -2r + (1 -r) 2 x -1 ] x 2 -2x(1 + r) + (1 -r) 2 + terms even under (s ↔ u) , (15)
with the abbreviations ρ ± = 1 + r ± 2 √ r. For the numerical evaluation of the (only relevant) real part of IV⊗A the spectral integral ∞ ρ + dx/(x -s) . . . in the second last line of eq.( 15) has to be treated as a principal value integral (if s > ρ + ). It can be conveniently decomposed into a sum of two non-singular integrals by the following master formula:

- ∞ ρ + dx f (x) x -s = 2s-ρ + ρ + dx f (x) -f (s) x -s + ∞ 2s-ρ + dx f (x) x -s . (16)
Finally, we are left with the contribution of the crossed two-photon exchange box diagram V shown in Fig. 4. Its interference term with the tree diagram can be obtained via crossing (s → u = 2 + 2r -s -t) as follows:

V ⊗ A = -IV ⊗ A | s↔u . (17)
The occurring minus-sign has the consequence that all terms even under the permutation of variables s ↔ u drop out in the sum (IV + V) ⊗ A and therefore we do not need to specify these. We have verified the relation eq.( 17) by an explicit calculation of V ⊗ A as it arises from the crossed two-photon exchange diagram V. The physical reason behind eq.( 17) can be easily explained and understood. When turning the right pion line in the crossed two-photon exchange diagram V upside-down one gets the planar two-photon exchange diagram IV for electron-antipion scattering (e -π + → e -π + ). The same manipulation applied to the tree diagram A introduces an additional minus sign due to the (single) opposite electric charge, and to close the argument, e -π + → e -π + and e -π -→ e -π -are connected with each other by s ↔ u crossing.

As an aside we note that it is instrumental to keep the factorized square roots in eqs.(5,10-15) as they stand. In this form the correct analytical continuation of the (only relevant) real parts of the loop functions along their branch cuts is guaranteed.

Infrared finiteness

In the next step we have to consider the infrared divergent terms proportional to ξ IR . Inspection of eqs. (5,10,15,17) reveals that these scale with the Born term A ⊗ A/t. As a consequence of that feature, the infrared divergent loop corrections multiply the differential cross section dσ/dΩ cm at tree-level by a factor:

δ (IR) virt = 4α π ξ IR 1 + t -2 √ 4 -t L(-t) + t -2r √ 4r 2 -rt L(-tr -1 ) +Re 2(1 + r -s) √ ρ + -s √ ρ --s ln √ ρ + -s + √ ρ --s 2r 1/4 + 2(u -1 -r) √ ρ + -u √ ρ --u ln √ ρ + -u + √ ρ --u 2r 1/4 . ( 18
)
The unphysical infrared divergence ξ IR gets removed in the measurable cross section by contributions from (undetected) soft photon bremsstrahlung. In its final effect, the (single) soft photon radiation off the in-or out-going electrons and pions yields the multiplicative factor:

δ soft = α µ 4-d | l |<λ d d-1 l (2π) d-2 l 0 2p 1 • p 3 p 1 • l p 3 • l + 2p 2 • p 4 p 2 • l p 4 • l + 2p 1 • p 4 p 1 • l p 4 • l + 2p 2 • p 3 p 2 • l p 3 • l - 2p 1 • p 2 p 1 • l p 2 • l - 2p 3 • p 4 p 3 • l p 4 • l - m 2 e (p 1 • l) 2 - m 2 π (p 2 • l) 2 - m 2 e (p 3 • l) 2 - m 2 π (p 4 • l) 2 , ( 19
)
which depends on a small photon energy cut-off λ. Working out this momentum space integral by the method of dimensional regularization (with d > 4) one obtains the following soft photon correction factor [11]:

δ (cm) soft = α π 4 1 + t -2 √ 4 -t L(-t) + t -2r √ 4r 2 -rt L - t r + 2 √ P (s -1 -r) ln √ s -ρ + + √ s -ρ - 2r 1/4 + 2(u -1 -r) √ ρ + -u √ ρ --u ln √ ρ + -u + √ ρ --u 2r 1/4 ln m e 2λ -ξ IR + 2 √ P × (s -1 + r) ln s -1 + r + √ P 2 √ sr + (s + 1 -r) ln s + 1 -r + √ P 2 √ s + 1/2 0 dx (t -2)(s + 1 -r) [1 -tx(1 -x)] √ R t ln s + 1 -r + √ R t s + 1 -r - √ R t + (t -2r)(s -1 + r) [r -tx(1 -x)] √ R t ln s -1 + r + √ R t s -1 + r - √ R t + 1 0 dx (s -1 -r)[s + (1 -r)(1 -2x)] (1 -2x)[sx(1 -x) + (1 -2x)(1 -x -rx)] √ P × ln s + (1 -2x)(1 -r + √ P ) s + (1 -2x)(1 -r - √ P ) + (u -1 -r)[s + (1 -r)(1 -2x)] [1 + (r -1)x -ux(1 -x)] √ R u ln s + (1 -r)(1 -2x) + √ R u s + (1 -r)(1 -2x) - √ R u , (20)
where we have introduced the polynomials:

P = s 2 -2s(1 +r)+(1-r) 2 , R t = P +4stx(1 -x), R u = P + 4x(1 -x)[su -(1 -r) 2 ]
. We note that the terms beyond those proportional to ln(m e /2λ) -ξ IR are specific for the evaluation of the soft photon correction factor δ soft in the center-of-mass frame with λ an infrared cut-off therein. As it is written in eq.( 20), δ

soft refers to an (idealized) experimental situation where all undetected soft photon radiation fills a small sphere of radius λ in the center-of-mass frame. In a real experiment the momentum space region of undetected photons can be of different (non-isotropic) shape with no sharp boundaries due to detector efficiencies etc. Such additional experiment-specific radiative corrections can be accounted for and calculated by integrating the fivefold differential cross section for e -π -→ e -π -γ over the appropriate region in phase space. By construction this region excludes the infrared singular domain | l | < λ and thus leads to a finite result. The treatment of hard photon bremsstrahlung [8] may also be important for comparison with a real experiment.

Numerical results

We are now in the position to present some numerical results for the radiative corrections to electron-pion scattering e -π -→ e -π -. The complete (infrared-finite) radiative correction factor is δ

(cm)
soft written in eq.( 20) plus the sum of all interference terms (see eq.( 2)) divided by the Born term A⊗A/t 2 . We refer to the center-of-mass kinematics where t = (cos θ cm -1)P/2s (and u = 2 + 2r -s -t), with θ cm the scattering angle and P = s 2 -2s(1 + r) + (1 -r) 2 . Fig. 5 shows in percent the radiative corrections arising from loops alone (discarding the ξ IR terms) at five selected center-of-mass energies √ s m π = (1, 1, 1.6, 2.1, 2.6, 3.1)m π as a function of cos θ cm . One notices sizeable negative radiative corrections with values up to about -11.5%. With increasing √ s the curves become gradually more symmetric under forward and backward directions, θ cm → 180 • -θ cm . At the highest selected center-of-mass energy √ s m π = 3.1m π the maximal squared momentum transfer is -t max m 2 π = 0.15 GeV 2 . This corresponds to the region within which accurate data for pion-electron scattering 2 are presently available [10]. 2 For elastic pion-electron scattering the squared momentum transfer is given by -tm 2 π = 2m e (p 0 3 -m e ) with p 0 3 the electron recoil energy. Since m e occurs as a determining factor, pion beams with energies in the Let us look also at individual contributions to the radiative corrections. For example, at √ s = 3 and θ cm = 120 • one has [(-14.90 -2.51) + (1.74 + 0.16 + 0.016) + 5.51]% = -9.99%, where the numbers in brackets stem from photonic vertex correction, vacuum polarization and two-photon-exchange, in the order presented in subsection 2.1. The contribution from the twophoton exchange triangle diagram III is in fact tiny: -0.0003%. This observation supports the approach of ref. [7] that the finite size of the pion can be ignored for this (extremely small) contribution. Evidently, the radiative corrections to electron-pion scattering e -π -→ e -π - are dominated by the electronic vertex correction due to the leading squared logarithm in the asymptotic behavior of the Dirac form factor F 1 (t) γ-loop -(α/4π) ln 2 (-tr -1 ). Next in importance follows the two-photon exchange in form of the box diagrams IV and V. In principle the contribution II π ⊗ A from pionic vacuum polarization should replaced by that of hadronic vacuum polarization (A ⊗ A)Π had (t)/t. However, for the small momentum transfers √ -tm π < 0.4 GeV considered here, the corresponding (hadronic) effect is very small and reasonably well approximated by the vacuum polarization due to a pointlike pion. The empirical vacuum polarization function with its leptonic and hadronic contributions is shown in Fig. 82 of ref. [START_REF] Actis | Working Group on Radiative Corrections and Monto Carlo Generators for Low Energies[END_REF]. Fig. 6 shows the radiative corrections to electron-pion scattering e -π -→ e -π -with inclusion of the soft photon bremsstrahlung effects. For the sake of having a concrete case we have set the infrared cutoff to the value λ = √ s m π /200, thus modelling an (idealized) experimental situation where the electron and pion energies can be resolved within 1% accuracy. One observes that the soft photon bremsstrahlung increases moderately the (negative) radiative corrections to electron-pion scattering. At √ s = 3 and θ cm = 120 • one has a soft photon correction of (10.50 -17.41)% = -6.91%, where the first and second number refer to the "universal" part proportional to ln(m e /2λ) and to the remaining terms in eq.( 20) specific for imposing an infrared cutoff via | l | < λ the center-of-mass frame. It is clear that the radiative few TeV range would be needed in order to reach a (modest) momentum transfer of √ -tm π 1 GeV. corrections due to the inelastic bremsstrahlung process e -π -→ e -π -γ depend crucially on the particular experimental conditions (i.e. cuts on energies and angles).

Radiative corrections to pion-pair production

In this section we discuss the radiative corrections to pion-pair production in e -e + annihilation.

The process e -e + → π + π -is obtained from elastic electron-pion scattering e -π -→ e -π -via the crossing transformation s ↔ t. We prefer to keep the original meaning of the Mandelstam variables and thus denote by √ t m π the total center-of-mass energy of the reaction e -e + → π + π -. The cosine of the scattering angle y = cos θ cm is involved in the (uncrossed) variable u = 1 + r + (y √ t -4 √ t -4r -t)/2 and s = 2 + 2r -t -u receives the same form with y → -y. The relevant angular average 1 2 1 -1 dy . . . of the tree amplitude A ⊗ A comes out as (4 -t)(t + 2r)/6. With an additional factor -1/2 due to averaging over positron spin-states the total cross section for pion-pair production has the following well-known form:

σ tot (t) = πα 2 3m 2 π t 3 (t -4) 3/2 √ t -4r (t + 2r) . ( 21
)
The analytical expressions for the loop contributions to pion-pair production e -e + → π + π - can be directly taken over from subsection 2.1. The angular integration 2π 1 -1 dy involved in the total cross section eliminates all terms which are odd under the permutation of variables s ↔ u. These are precisely the two-photon exchange contributions (III+IV+V)⊗A given in eqs. (13,15,17). The virtual radiative corrections (of order α) to the total cross section σ tot (e -e + → π + π -) are therefore completely determined by the photonic vertex corrections I⊗A and the vacuum polarizations II⊗A. By making use of the angular average (4-t)(t+2r)/6 of A ⊗ A they can be directly read off from eqs. (5,(10)(11)(12) with their explicit t-dependence.

The soft photon correction factor for pion-pair production e -e + → π + π -is a modified version of eq.( 19) due to sign-changes of charges. When evaluated for the center-of-mass kinematics under consideration it reads [11]:

δ (cm) soft = α π 4 1 + 2 -t √ t L(t -4) + 2r -t √ tr L t r -4 ln m e 2λ -ξ IR +2 t r L t r -4 + 2 √ t L(t -4) + 1/2 0 dx √ t(2 -t)(t -4) -1/2 (1 -2x)[1 + (t -4)x(1 -x)] ln √ t + (1 -2x) √ t -4 √ t -(1 -2x) √ t -4 + √ t(2r -t)(t -4r) -1/2 (1 -2x)[r + (t -4r)x(1 -x)] ln √ t + (1 -2x) √ t -4r √ t -(1 -2x) √ t -4r + terms odd under (s ↔ u) , (22)
with the function L(x) defined in eq.( 6). The terms odd under s ↔ u need not to be specified. After multiplication with the even Born term A ⊗ A and subsequent angular integration 2π 1 -1 dy they drop out of the radiative corrections to σ tot (t). The dashed line in Fig. 7 shows the radiative corrections to the total cross section for pion-pair production e -e + → π + π -arising from loops alone (discarding the ξ IR terms) . One observes sizeable (negative) radiative corrections which increase with center-off-mass energy √ t m π . For example, at √ t = 4 one has [(-16.80 -2.70) + (1.91 + 0.22)]% = -17.37%, where the numbers in brackets correspond to photonic vertex correction and leptonic vacuum polarization, in the order presented in subsection 2.1. The photon-loop correction to the photon-electron vertex is again the dominant effect. Next in importance follows the same type of correction to the photon-pion vertex. The dashed line in Fig. 8 shows the radiative corrections to the total cross section for pion-pair production e -e + → π + π -with inclusion of soft photon bremsstrahlung effects. For the sake of having a concrete case we have set the infrared cutoff to the value λ = √ t m π /200. One observes a further increase of the negative radiative corrections, but now their rise with the center-of-mass energy

√

t m π has significantly slowed down. Note that we have dropped here the contribution II π ⊗ A from pionic vacuum polarization since it does not even in an approximate way represent the effects from hadronic vacuum polarization in the energy region √ tm π < 1 GeV of interest. As can be seen from Fig. 82 in ref. [START_REF] Actis | Working Group on Radiative Corrections and Monto Carlo Generators for Low Energies[END_REF] the empirical hadronic vacuum polarization function develops pronounced oscillations with sharp spikes in that energy range. Radiative corrections to the differential cross section for e -e + → π + π -at yet higher orders in α have been studied by Hoefer et al. in ref. [12].

Inclusion of pion structure

With good reason one may object that treating the pion as a structureless spin-0 particle in a calculation of the radiative corrections to e -e + → π + π -for energies up to √ t m π 1 GeV is rather unrealistic. In the ρ(770)-resonance region the empirical charge form factor of the pion leads to an enhancement of the total cross section by a factor of up to 45. One is therefore obliged to include the structure (finite size) of the pion also in the calculation of the radiative corrections. As motivated by the vector meson dominance model, we introduce at each photonpion vertex a "bare" pion charge form factor of monopole type: F π (-q 2 ) = (1 -q 2 /m 2 ρ ) -1 , with m ρ = 770 MeV the (neutral) ρ-meson mass. Here, q is the photon four-momentum and Radiative corrections to the total cross section for pion-pair production e -e + → π + π -arising from loops. The dashed line corresponds to a structureless pion. The full line includes pion-structure through a vector meson dominance form factor. Hadronic vacuum polarization [START_REF] Actis | Working Group on Radiative Corrections and Monto Carlo Generators for Low Energies[END_REF] is not considered explicitly. Radiative corrections to the total cross section for pion-pair production e -e + → π + π -with inclusion of soft photon bremsstrahlung. The dashed line corresponds to a structureless pion. The full line includes pion-structure through a vector meson dominance form factor. Hadronic vacuum polarization [START_REF] Actis | Working Group on Radiative Corrections and Monto Carlo Generators for Low Energies[END_REF] is not considered explicitly.

we apply the same form of F π (-q 2 ) irrespective of whether the in-and out-going pions are on or off their mass-shell. A first success of such an (admittedly crude) approach is the pion electromagnetic selfenergy. When evaluated with a squared monopole form factor inserted the one-photon loop selfenergy diagrams lead to the following expression for the mass difference between the charged and neutral pion:

∆ π = α 2π m ρ z + 2z 3 ln 2z -(2z 2 + 1) √ z 2 -1 ln(z + √ z 2 -1) , (23)
with the ratio z = m ρ /2m π 0 = 2.852. The value ∆ π = 4.31 MeV resulting from this simple model is remarkably close to the experimental pion mass difference ∆ (exp) π = 4.59 MeV. The small remainder is (generally) attributed to an isospin-breaking effect quadratic in the upand down-quark mass difference.

We apply now the monopole form factor F π (-q 2 ) consistently to all photon-pion vertices of the diagrams in Fig. 1,2,3 and mark these modified diagrams by a prime. The modified Born term reads:

A ⊗ A = b 2 (b -t) 2 (A ⊗ A) , (24)
with b = (m ρ /m π ) 2 = 30.44, and similarly one gets for diagram I e with electronic vertex correction:

I e ⊗ A = b 2 (b -t) 2 (I e ⊗ A) . (25)
For the (pionic) vertex correction diagrams I π we exploit identity:

m 4 ρ (-l 2 )(m 2 ρ -l 2 ) 2 = 1 -l 2 + m 2 ρ ∂ ∂m 2 ρ -1 1 m 2 ρ -l 2 , (26)
which shows that the photon-loop with a squared monopole form factor inserted is equal to an ordinary photon-loop plus a term which can be calculated from a massive vector-meson loop by differentiation with respect to its mass. It is most convenient to calculate directly the imaginary part of the (ordinary) vector-meson loop via the Cutkosky cutting rule and to apply (after the mass-differentiation) a once-subtracted dispersion relation. 3 Putting all the pieces together one obtains the following modified interference term:

I π ⊗ A = b 2 (b -t) 2 (I π ⊗ A) + α 2π (A ⊗ A ) ∞ 4 dx 1 √ x 2 -4x (x -t) × 1 - 2 + b x - b 2(x -4 + b) + 2 x -1 + b 2 x 2 -4x ln x -4 + b b . (27)
For t > 4, the real part of the spectral integral ∞ 4 dx/(x -t) . . . is given by its principal value. Finally, there are the modified contributions from electronic and muonic vacuum polarization:

II e,µ ⊗ A = b 2 (b -t) 2 (II e,µ ⊗ A) , (28)
as well as from hadronic vacuum polarization:

II had ⊗ A = A ⊗ A t Π had (t) , Re Π had (t) = t 4π 2 α - ∞ 4m 2 π ds σ 0 had (s ) t -s m -2 π , (29)
with σ 0 had (s ) the (undressed) cross section for electron-positron annihilation into hadrons [START_REF] Actis | Working Group on Radiative Corrections and Monto Carlo Generators for Low Energies[END_REF]. The two-photon exchange contribution (III + IV + V )⊗A remains of course odd under s ↔ u and thus drops out of the radiative corrections to the total cross section σ tot (e -e + → π + π -). Note that the assumption of a monopole for the pion charge form factor plays only a role for the second term in eq.( 27). After dividing by the modified Born term A ⊗ A the actual functional form of |F π (-tm 2 π )| 2 drops out in the radiative corrections of interest. Accurate empirical parameterization of the pion charge form factor F π (-q 2 ) which include the finite width of the (neutral) ρ(770)-meson and the (isospin-breaking) ρ 0 -ω interference can be found in appendix C of ref. [12].

The full line in Fig. 7 shows the radiative corrections to σ tot (e -e + → π + π -) arising from loops with inclusion of the pion-structure. The individual contributions to the radiative correction factor at √ t = 4 are now: [(-16.80 -2.83) + (1.91 + 0.22)]% = -17.50%. By comparison with the dashed line in Fig. 7 one sees that the effects due to the pion-structure are very small. The essentially new principal-value integral term in eq.( 27) arising from the photonloop with monopole-type vertex functions amounts at √ t = 4, 5.5, 7 to radiative corrections of (-0.13, -0.20, -0.25)%, respectively. The full line in Fig. 8 shows the analogous results with inclusion of the soft photon bremsstrahlung effects. The latter are given by eq.( 22) in unchanged form since for real photon emission (q 2 = 0) the form factor effect vanishes (according to our treatment which does not make a distinction between on-and off-shell pions).

Let us exhibit separately the correction induced by the finite size of the pion on the photonloop form factor G π (t) γ-loop of the pion:

δG π (t) γ-loop = αt 2π ∞ 4 dx x -t -i0 1 √ x 2 -4x 1 - b 2(x -4 + b) - 2 + b x + 2 x -1 + b 2 x 2 -4x ln x -4 + b b . (30)
The full and dashed line in Fig. 9 show the real and imaginary part of this quantity in the interval -50 < t < 50. One observes very small values which range between 1.1 permille and -1.4 permille. At the branch point t = 4 this finite size correction reads:

δG π (4) γ-loop = α 6π 3 + 2b -b 2 ln b + 2 b(b -4) (b 3 -2b 2 -2b -6) ln √ b + √ b -4 2 , (31)
and the slope at t = 0 is given by:

δG π (0) γ-loop = α 24π (4 + b 2 ) ln b -3 -2b - 20 b -4 + 2 √ b (b -4) 3/2 ×(6b 2 -b 3 -10b + 18) ln √ b + √ b -4 2 , (32)
with b = (m ρ /m π ) 2 = 30.44. It remains to be seen how much the finite size correction δG π (t) γ-loop changes if an improved parameterization of the half off-shell pion-photon vertex function is used. The electromagnetic selfenergy ∆ π given in eq.( 23) suggests that the present treatment could be fairly reasonable. Finally, we note that the radiative corrections to pion-and kaon-pair production in e -e + collisions for energies below 2 GeV have also been calculated by Arbuzov et al. in ref. [13]. In that work the differential cross section dσ/dΩ cm of the reaction has been considered and the radiative corrections have been approximated by their leading logarithms in the electron mass m e . The structure (i.e. finite size) of the mesons has however not been taken into account and the accuracy of their calculation [13] has been estimated to about 0.2%. This numerical estimate is remarkably consistent with our present finding for the size of the additional radiative corrections induced by the pion-structure.

Figure 2 :

 2 Figure 2: One-loop diagrams (I) with photonic vertex corrections. The right diagram has a horizontally reflected partner.

Figure 3 :

 3 Figure 3: One-loop diagrams (II) with leptonic or pionic vacuum polarizations.

Figure 4 :

 4 Figure 4: Two-photon exchange diagrams (III), (IV) and (V) for electron-pion scattering.

Figure 5 :

 5 Figure 5: Radiative corrections to electron-pion scattering e -π -→ e -π -arising from loops. The numbers (1.1, 1.6, 2.1, 2.6, 3.1) on the curves correspond to the total center-of-mass energy √ s m π in units of m π .

Figure 6 :

 6 Figure 6: Radiative corrections to electron-pion scattering e -π -→ e -π -including soft photon bremsstrahlung. The numbers (1.1, 1.6, 2.1, 2.6, 3.1) on the curves correspond to the total center-of-mass energy √ s m π in units of m π . The infrared cutoff has been set to λ = √ s m π /200.

Figure 7 :

 7 Figure7: Radiative corrections to the total cross section for pion-pair production e -e + → π + π -arising from loops. The dashed line corresponds to a structureless pion. The full line includes pion-structure through a vector meson dominance form factor. Hadronic vacuum polarization[START_REF] Actis | Working Group on Radiative Corrections and Monto Carlo Generators for Low Energies[END_REF] is not considered explicitly.

Figure 8 :

 8 Figure8: Radiative corrections to the total cross section for pion-pair production e -e + → π + π -with inclusion of soft photon bremsstrahlung. The dashed line corresponds to a structureless pion. The full line includes pion-structure through a vector meson dominance form factor. Hadronic vacuum polarization[START_REF] Actis | Working Group on Radiative Corrections and Monto Carlo Generators for Low Energies[END_REF] is not considered explicitly.

Figure 9 :

 9 Figure 9: Correction to the photon-loop form factor of the pion induces by the finite size of the pion. The calculation is based on a pion charge from factor of monopole type.

If an infinitesimal photon mass m γ is introduced as an (alternative) infrared regulator the infrared divergence ξ IR is to be identified with the logarithm ln(m e /m γ ).

We have verified that the normalization condition G π (0) γ-loop = 0 for the photon-loop induced pion form factor is preserved if the vertices include a monopole form factor. In addition to the diagrams I π in Fig.2there is also a b-dependent pion wavefunction renormalization factor, Z (π) 2 -1 = (α/4π)

4ξ IR -2 ln r + b -2 + (2b 2 /2) ln b + b/(b -4) (b 2 -2b -2) ln[( √ b + √ b -4)/2], which contributes to the zero-sum.