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Knockout  reactions on nuclei


In knockout  reactions the quantum numbers of the emitted nucleon (and, consequently, of the residual nucleus) are known. The reaction is completely exclusive and  detailed information on the spectroscopic structure of the target nucleus and on the nucleon dynamics inside the hadronic matter can be obtained, since the electromagnetic probe is able to explore the whole nuclear volume.

In this context, I've been working with the theoretical group in Nuclear Physics at the University of Pavia to implement a model for exploring the limits of an effective description based on a single-particle dynamics. In particular, I studied the two-body correlation effects in the electromagnetic interaction between the external probe and the bounded nucleon (publ. a2, a3-a5, a8, a9) and in the residual interaction between the emitted nucleon and the surrounding nuclear medium (publ. a1, a6), using also the socalled correlated Glauber method in the framework of a collaboration with the theoretical group of prof. V.R. Pandharipande at the University of Illinois at Urbana-Champaign (USA) (publ. a11). I developed a suited FORTRAN code (publ. d2) that has been adopted in several laboratories (NIKHEF - Amsterdam, SACLAY - Francia, TJNAF - USA, MIT - USA) to analyze experimental data. This represented the ground for a proposal of experiment at the  MIT-Bates  laboratory (publ. a7, d3) and the analyzing tool for the first ever measured asymmetry of the nuclear response with respect to the scattering plane in a knockout  reaction with an incident polarized electron beam  (publ. a10, a21).

I started a collaboration between the Pavia group and the  Washington University at St.Louis (USA) to implement the model with the spectral density formalism. The microscopic description of dynamics now contains realistic particole-hole correlations that reproduce the complicated spectroscopic structure of the experimentally observed excitation levels for the residual nucleus (publ. a15). More recently, I worked to extend the applicability of the model to the high-momentum transfer regime (beyond the inelastic threshold of about 1 GeV/c), where relativistic effects are relevant. It was then possible,  starting from a microscopic description of the dynamics based on a hole spectral density for the bounded state and on a particle spectral density for the continuum state, to give for the first time a coherent picture of the spectroscopic properties of the 16O nucleus reproducing experimental data obtained in different times, laboratories, and, in particular, at very different  energy scales (publ a29, a30).

When the momentum transfer exceeds the above mentioned inelastic threshold, according to QCD new colorless  hadronic small configurations in transverse space are predicted that should propagate and form hadrons with a strongly reduced interaction with the surrounding nuclear medium. This nonperturbative QCD prediction, known as color transparency, is beng verified in dedicated experiments in several international laboratories. In collaboration with A. Bianconi from  University of Brescia, I analyzed the approximations that are usually assumed in the literature to describe the scattering wave function of the emitted nucleon (publ. a12, a13, a17), deducing a working criterium to identify nuclear transparency effects (publ. a14) and giving an alternative interpretation of some experimental data in terms of superposition of nuclear structure effects (publ. a18).

Finally, in collaboration with S. Boffi, C. Giusti and F.D. Pacati from University of Pavia, by invitation of P.E. Hodgson, General Editor of Oxford University Press, I wrote a book on the electromagnetic response of the nucleus, published in the series Oxford Studies in Nuclear Physic (publ. c1). In particular, I took care of the chapters  about the semi-inclusive and exclusive reactions with emission of one nucleon at low, medium and high energies, and the extension of the formalism and the phenomenology to the polarized reactions. At the time of its publication (1996), the book represented a turning point, summarizing the acquired knowledge in this field in a homogeneous formalism, but at the same time anticipating the topics and developments that should be achieved in the following years in higher-energy experiments (in particular, at the TJNAF laboratory, Virginia - USA). Because of the modern approach, the uniform formalism, and the wide variety of applications, the book has been adopted in Ph. D. courses in Physics in several american universities. Moreover, it is largely cited in research papers (see  Google-Scholar), because, at present, it represents the only theoretical reference book for the experimental researchers in this field.
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Quark models of the nucleon


When the QCD chiral symmetry is spontaneously broken, new dynamical degrees of freedom come into play: quarks get a dynamical mass related to the nontrivial structure of the ground state (constituent quarks), the Goldstone mechanism produces new bosons (usually identified with the pseudoscalar mesonic octet) that mediate the interactions among quarks. Traditional constituent quark models do not include the latter feature, since they describe the dynamics mainly by  the hyperfine part of the one-gluon exchange interaction. There are "hybrid" versions that include also the one-pion exchange interaction, but the short-range behaviour and the corresponding spin-flavor symmetry do not seem adequate to reproduce the fine structure of the baryon spectrum, in particular the correct series of positive- and/or negative-parity energy levels.

In this context, I first studied the predictions of these models about the electromagnetic excitation of the most important nucleon resonances, namely the Delta(1232), the D13(1520) and the F15(1680), testing the limit of such framework when trying to coherently describe experimental data for the corresponding helicity amplitudes (publ. a16).

Next, in collaboration with the group of W. Plessas from the University of Graz, I implemented a new model where constituent quarks interact via the exchange of just the pseudoscalar meson octet, which is justified as the set of Goldstone bosons produced by the spontaneously broken QCD chiral symmetry. Wave functions have been constrained by reproducing the spectrum of light baryons including strangeness, and then they have been tested by calculating dynamical properties of the baryons themselves (form factors, magnetic moments,..) using first a semirelativistic description of the electromagnetic interaction (publ. a19, a20, a24). Later, in collaboration also with W. Klink from Iowa University (USA), a covariant hamiltonian description was introduced  (using the point-form realization of relativistic quantum mechanics). This way, it was possible to simultaneously describe for the first time  all the elastic form factors of the nucleon (electric, GE, magnetic, GM, axial, GA, pseudoscalar, GP, and, in particular, the last data collected at  TJNAF - USA - about the ratio GE/GM for the proton), magnetic moments, root mean squared radii, without introducing new parameters with respect to the ones fixed in the Hamiltonian by reproducing the baryon spectrum (publ. a25-a27). Such a coherent picture strongly depends on the correct treatment of the relativistic aspects of the process. In the point-form realization of the relativistic quantum mechanics, the invariance under Poincaré transformations is trivial and, at variance with other approaches, it is possible to exactly calculate the transformations induced by the change of reference frame (boost).

The surprising behaviour of the GE/GM ratio for the proton, as measured at TJNAF-USA, stimulated a large theoretical activity, in particular about these form factors in the time-like region. In fact, while space-like form factors of stable hadrons are real because of the hermiticity of the electromagnetic Hamiltonian, time-like form factors, as they can be explored in electron-positron annihilations or hadronic collisions, are complex because of the (final/initial) residual interactions of the involved hadrons. The experimental knowledge of time-like form factors for the nucleon is poor. In principle, their absolute values can be extracted by combining the measurement of total cross sections and center-of-mass (c.m.) angular distributions of the final products; the phases can be deduced from specific spin asymmetries of the corresponding polarized reaction. In reality, the available unpolarized cross sections are integrated over a wide angular range because of low statistics with the net outcome that the absolute value of the proton electric form factor is basically unknown. Moreover, results from only one experiment are available for the neutron. Finally, no polarization data have ever been collected; therefore, also the phases of nucleon form factors are totally unknown, which could strongly discriminate the analytic continuation of models that successfully reproduce the ratio measured at JLab in the space-like region. Nevertheless, the few available results display puzzling properties. The recent BABAR measurement shows a proton electric-to-magnetic ratio larger than 1, which contradicts the space-like results of JLab and older time-like results of LEAR. The only neutron measurement from FENICE displays an absolute value of the magnetic form factor bigger than the proton one in the corresponding c.m. range. Similarly, the asymptotic behavior of the latter seems to contradict the requirements of dispersion relations, which would bind it to the corresponding trend in the space-like region. In collaboration with prof. A. Bianconi from University of Brescia, and with d.ssa B. Pasquini from University of Pavia, numerical simulations have been performed to explore the feasibility of the extraction of proton/neutron time-like form factors (publ. a42, a43) with the possible upgrade of the existing DAFNE facility to center-of-mass energies up to 2.5 GeV (see the Roadmap INFN 2006-2016, publ. a44, d6).

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Partonic (spin) structure of the nucleon and azimuthal asymmetries


The leading-order partonic structure of the nucleon is completely determined by three distribution functions: the momentum distribution, the helicity and the transverse spin distribution (transversity). Only the first two ones are experimentally known because the third one is odd under chiral transformations and it is not accessible in simple processes like inclusive DIS. However, knowledge of the transversity is crucial to determine the spin structure of the nucleon and to test QCD predictions in the nonperturbative regime about the tensor charge and its evolution properties with respect to the helicity ones. The strategy for extracting the transversity is simply defined: we need to identify a leading-order (-twist) process where the transversity is paired with a partner being itself also odd under chiral transformations.

The most natural process is the Drell-Yan with two transversely polarized protons, where at  leading twist the cross section contains the product of the transversities for the annihilating quark and antiquark. However, the antiquark transversity is evidently suppressed since it is not a valence distribution inside the parent proton.  For these two reasons, the suggestion to extract transversity from a fully polarized Drell-Yan appeared early in the literature, but it has been quickly discarded as well. Recently, new perspectives were offered in hadronic collisions by the developments of know-how about dealing with (un)polarized antiprotons inside the project FAIR (Facility with Antiproton and Ion Research) using the ring HESR (High Energy Storage Ring) at the  GSI laboratory (Darmstadt - Germany). In collaboration with prof. A. Bianconi from University of Brescia, we wrote a Monte Carlo code for numerical simulations of the $p^\uparrow \bar{p}^\uparrow \to \mu^+\mu^- X$ process in the GSI kinematics to explore the feasibility for extracting transversity (pubbl. a39).

If the elementary annihilation is assumed to be non collinear, i.e. the partonic densities depend on an intrinsic transverse momentum of the partons with respect to the parent hadrons, the leading-twist Drell-Yan cross section shows a rich azimuthal dependence. In fact, the transversity appears convolved with a new parton spin density, the socalled Boer-Mulders function, which in turn produces an azimuthally asymmetric term in the unpolarized cross section. Recently, the latter raised much interest since it could give an explanation for a long lasting puzzle that perturbative QCD has not solved yet (the socalled violation of the Lam-Tung sum rule). In the polarized part of the cross section, another contribution shows up involving another interesting spin density, the Sivers function. It describes how the distribution of unpolarized partons is distorted by the transverse polarization of the parent hadron; hence, it gives information on the orbital angular momentum of partons and on its contribution to the proton spin sum rule. Moreover, the general properties of the defining operator imply an anomalous feature of the Sivers function with respect to universality: as it can be extracted in Drell-Yan processes, it would come out opposite from what can be extracted in semi-inclusive DIS processes (SIDIS). In this context, I made numerical simulations at GSI kinematics to study the feasibility for extracting the transversity and Boer-Mulders function in Drell-Yan processes with unpolarized or a single transversely polarized proton (pubbl. a37), in the first case analyzing also the azimuthal asymmetries produced by the crossing of the target nuclear medium (pubbl. a38). I extended the analysis to the kinematics foreseen for the upgrade RHIC-II at the Brookhaven National Laboratory, in order to verify the feasibility of extracting the Sivers function and to test its universality properties in high-energy proton collisions (pubbl. a40). I studied also the same physics case for a possible configuration of the COMPASS experiment with pion beams in the $\pi p^\uparrow \to \mu^+\mu^- X$ reaction (pubbl. a41). Finally, I explored also the role of unpolarized nonvalence partons in single-spin asymmetries, using Monte Carlo simulations as well (pubbl. a46).



An alternative way to extract the transversity is to search for a semi-inclusive reaction in which the fragmentation function for the observed hadron is also odd under chiral transformations and it represents a partner for the transversity in the cross section at leading twist.  The new unknown function could be determined by looking at the corresponding semi-inclusive e+e- annihilation provided that the related factorization theorem holds and the fragmentation function is universal. All this has been realized for the first time only recently, by combining data from HERMES (DESY) for the $e p^\uparrow \to e' \pi X$ reaction and those from BELLE (KEK) for the $e^+e^- \to \pi^+ \pi^- X$ one. The unknown fragmentation function is called Collins function. However, the azimuthal asymmetry necessary to isolate the "Collins effect" demands the cross section to depend upon the transverse momentum of the observed pion, hence, at the elementary level, upon the intrinsic transverse momentum of the fragmenting parton. At present, this requirement prevents from getting a complete proof of the factorization theorem. Moreover, it affects the final result since the evolution of the transverse-momentum-dependent Collins function from the BELLE scale down to the HERMES one, is not known.

In this context, I studied an alternative SIDIS process where two hadrons are detected inside the same jet, looking at the general properties of the socalled interference fragmentation functions (or Dihadron Fragmentation Functions - DiFF) that show up at leading  (publ. a22, a32) and subleading twist (publ. a34). Surprisingly, it is more useful to consider a more complicate final state because it is possible to build a spin asymmetry that isolates transversity at leading twist without keeping memory of the parton transverse momenta (publ. a28).  The DiFF can be extracted from the corresponding  e+e- annihilation into two pion pairs, as their universality has been directly checked at leading twist (publ. a33), and their evolution equations have been determined at leading log approximation such that the e+e- cross section can be expressed in factorized form at the same level of accuracy (pubbl. a47).

Coming back shortly to the case for hadron-hadron collisions, an interesting use of DiFF is possible in azimuthal asymmetries in processes like $H_1 H_2^\uparrow \to (\pi^+ \pi^-) X$ and $H_1 H_2\to (\pi^+ \pi^-)(\pi^+ \pi^-) X$. In fact, it is possible to determine all the unknowns, namely the transversity in $H_2$ and the corresponding DiFF partner, and the unpolarized DiFF paired to the partonic momentum distribution in $H_1$ as well, by performing one experiment only and measuring in turn one or two pion pairs (pubbl. a36) . This idea is being studied by the COMPASS collaboration for the case $H_1 = \pi$ (pubbl. a41).


The scarce amount of experimental data and, consequently, the lacking of reliable parametrizations makes the use of phenomenological models of fragmentation functions unavoidable, but also useful to determine the experimental set-up. In the framework of the spectator model, I co-worked an explorative calculation of interference fragmentation functions for the production of a nucleon and a pion either directly or through the Roper resonance (publ. a23), and for the production of two pions either directly in relative s  wave or through the $\rho$ resonance in p wave (publ. a28). Recently, the model has been enlarged to include more resonances and, more importantly, by constraining the free parameters to the invariant-mass distribution of the pair, as it is output by the Monte Carlo PYTHIA code of the HERMES collaboration (pubbl. a45). In this way, it was possible to predict the single-spin asymmetry from which transversity can be extracted. The feasibility of azimuthal asymmetry measurements containing such objects has been demonstrated by the collaborations HERMES (DESY) and COMPASS (CERN). Future upgrades of such a model require a more careful analysis of the statistics of the two pions in the final state, and the support to the analysis of the $e^+e^- \to (\pi^+\pi-)(\pi^+\pi^-) X$ reaction from the BELLE collaboration, in order to extract for the first time the chiral-odd DiFF that is the transversity partner.

The quest for understanding the spin structure of the nucleon has benefit from the development of other sophisticated tools like the Generalized Parton Distributions (GPD), that are partonic correlation functions based on nondiagonal hadronic matrix elements. As such, they include as a specific limit the usual parton distributions; but specific GPD moments are related to the electromagnetic form factors in the exclusive limit. More generally, GPD represent a unifying formalism for different classes of inclusive and exclusive processes. Moreover, in the nucleon rest frame GPD produce a 3dim map of the strong force, hence they contain information on the 3dim (and transverse) localization of partons and on their orbital angular momentum, giving for the first time the opportunity to gain a complete and  gauge invariant information on the partonic spin structure of the nucleon.

GPD can be extracted from  (spin or beam charge) asymmetry measurements in reactions like the Deeply Virtual Compton Scattering (DVCS) or the exclusive electroproduction of mesons. However, the complete determination of GPD is still an open problem. Therefore, the use of phenomenological models is still the only available tool, at present. In collaboration with the group of K. Goeke from the  University of Bochum , I checked that the chiral quark soliton model of the nucleon, based on the instanton description of the QCD vacuum, meets some crucial properties of GPD that are derived from general principles and, therefore, must be satisfied by any model. In particular, the chiral quark soliton model meets the consistency check of the socalled polinomiality condition which derives from hermiticity and invariance under Lorentz, parity and time-reversal transformations  (publ. a31).

In this framework, from the collaboration of several active european research groups the project Integrated Infrastructure Initiative in Hadronic Physics (I3HP) was elaborated, submitted and funded by the European Community under the contract n. RII3-CT-2004-506078 inside the programme FP6 . The I3HP project contains several activities; I take part in three of them: