aa r X i v : . [ nu c l - t h ] J a n High Energy Break-Up of Few-Nucleon Systems
Misak SargsianDepartment of Physics, Florida International University, Miami, FL 33199November 1, 2018
AbstractAbstract
We discus recent developments in theory of high energy two-body break-up reactions of few-nucleon systems. The characteristics of these reactions are such thatthe hard two-body quasielastic subprocess can be clearly separated from the accompa-nying soft subprocesses. We discuss in details the hard rescattering model (HRM) inwhich hard photodisintegration develops in two stages. At first, photon knocks-out anenergetic quark which rescatters subsequently with a quark of the other nucleon. Thelatter provides a mechanism of sharing the initial high momentum of the photon bythe outgoing two nucleons. Within HRM we discuss hard break-up reactions involving D and He targets. Another development of HRM is the prediction of new helicityselection mechanism for hard two-body reactions, which was apparently confirmed inthe recent JLab experiment. There is class of high-energy and momentum transfer (semi) exclusive nuclear reactionswhich suits very well for studies of Nuclear QCD. There are several common features intheoretical approaches for studying these reactions: (a) one can clearly identify a hard sub-process; (b) which can be factorized from the soft nuclear part of the reaction; (c) thesubprocess is hard enough for pQCD to be applicable; (d) soft part of the reaction couldbe expressed through the measurable/extractable quantities such as partonic distributionfunctions, hadron-hadron scattering amplitudes, form-factors and calculable nuclear wavefunctions. If all above requirements are achieved then such theoretical approaches may yield(sometimes) parameter free predictions (see e.g. [1]) and practically always will allow us tostudy the aspects of strong interaction dynamics which otherwise can not be investigatedwithout using nuclear targets.Several of such reactions which satisfy above criteria are, (i) semiinclusive deep-inelasticnuclear reactions aimed at studies of nuclear modifications of PDFs (EMC effects)(see e.g.[2]); (ii) high momentum transfer elastic scattering off few-nucleon systems that can beused to study the onset of quark degrees of freedom in strongly correlated few nucleonsystems[3], (iii) electrodisintegration of few-nucleon systems in the kinematics dominated byfinal state interaction that is used to probe the dominance of point like configurations in thehadronic wave function at large Q (color transparency/coherence phenomena) [4, 5, 6, 7];1 c) N N(a) (b)
Figure 1: Possible QCD dynamics of NN interaction.(iv) DIS nuclear scattering at x Bj > In this presentation we focus on the last class (v) of the reactions, in which high energyphoton produces two energetic nucleons which equally share the initial energy of the photon.These reaction kinematically corresponds to the break-up of 2N system at 90 angle in the γ – 2 N center of mass reference frame.Due to completely symmetric configuration, photon predominantly probes the structureand the dynamics of the exchanged particle in the NN system (Fig.1). High energy ofphoton in this case provides necessary resolution to probe the QCD content of NN interactionwhether it proceeds through the q ¯ q exchange (Fig.1b), quark interchange (Fig.1c) or gluonexchange (Fig.1d).Effectiveness of these processes in probing QCD aspects of nuclear interaction can beseen (for γd → pn reactions) from the following kinematical considerations[3, 9, 10]) inwhich: s = ( k γ + p d ) = 2 M d E γ + M d ; t = ( k γ − p N ) = ( cos ( θ cm ) − s − M d . (1)Simple estimate shows that already at E γ > − t | > and invariant mass of the N N system M NN = √ s > γd → pn reactions was the prediction of the s dependence of the differential cross section based onthe quark counting rule[3], which yields: dσdt ∼ s − . This prediction was experimentallyconfirmed already starting at E γ = 1 GeV for several set of experiments at SLAC[12, 13]and Jefferson Lab[14, 15, 16, 17].The quark counting predictions are based on the hypothesis that the Fock states withminimal number of partonic constituents dominate in two-body large angle hard collisions[18].Although successful in describing energy dependences of the number of hard processes, thishypothesis does not allow to make calculation of the absolute values of cross sections. Espe-cially for reactions involving baryons, calculations within perturbative QCD underestimate2 BA pp q kk ppp d Figure 2: Typical diagram for hard rescattering mechanism.the measured cross sections by orders of magnitude (see e.g.[19]). This may be an indi-cation that in the accessible range of energies bulk of the interaction is in the domain ofnonperturbative QCD[19, 20]. However, the main problem is that even if we fully realize theimportance of nonperturbative interactions the theoretical methods of calculations in thenonperturbative domain are very restricted.
The underlying assumption in hard rescattering model (HRM)[1] is that high energy photo-disintegration of two-nucleon system proceeds through the two stages in which an absorptionof photon by a quark of one nucleon is followed by a high-momentum transfer (hard) rescat-tering with a quark from the second nucleon. The latter rescattering produces a final twonucleon state with large relative momenta. A typical diagram representing such a scenariois presented in Fig.2.Analyzing the type of diagrams as in Fig.2 allows us to do the following observations: • the dominant contribution comes from the soft vertices of d → N N transition, whilequark rescattering proceeds trough the hard gluon exchange, • the d → N N transition can be evaluated through the conventional deuteron wavefunctions, • the structure of hard quark interchange interaction in the rescattering part of thereaction is similar to that of hard NN scattering, • as a result the sum of the multitude of diagrams with incalculable nonperturbative partof the interaction can be expressed through the experimentally measured amplitude ofhard N N scattering.Based on these observations, calculation of the γ + d → pn amplitude yields[1, 21] h p λ A , n λ B | A | λ γ , λ D i = X λ f ( θ cm )3 √ s ′ Z Ψ λ D ,λ γ ,λ ( α c , p ⊥ ) d p ⊥ (2 π ) × (cid:16) h p λ A , n λ B | A pn ( s, t n ) | p λ γ , n λ i− h p λ A , n λ B | A pn ( s, u n ) | n λ γ p λ i (cid:17) , (2)3 .10.20.30.40.50.60.70.80.91 2 2.25 2.5 2.75 3 3.25 3.5 3.75 4 4.25 4.5 E γ , GeV s d σ / d t ( kb - G e V ) Θ cm , deg d σ / d Ω c m ( nb / s r ) γ + d → p + n γ + d → p + n, E γ =2.4 GeV Figure 3: Energy dependence of the scaled cross section at 90 CM scattering (top) andangular dependence of the cross section at E γ = 2 . A pn is high momentum transfer elastic pn scattering amplitude, | N λ i ( N = p, n )represents the helicity wave function of nucleon and λ γ is the helicity of incoming photon.Based on Eq.(2) one obtains the following expression for the differential cross section of γ + d → pn reaction: dσ γd → pn dt = 8 α π s ′ C ( ˜ ts ) dσ pn → pn dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z Ψ NRd ( p z = 0 , p t ) √ m n d p t (2 π ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (3)where s ′ = s − m N and ˜ t = ( p n − m n ) . The interesting properly of the function C isthat C ( θ cm = 90 ) ≈
1. Therefore for 90 CM scattering HRM prediction is parameter free.Since pn cross section in high momentum transfer behaves like s − , Eq.(3) predicts same s − dependence as quark counting rule without requiring an onset of pQCD regime. Also, dueto angular dependence of the pn amplitude, HRM predicts an angular distribution being notsymmetric around 90 CM. These predictions agree reasonably well with the experimentaldata[14, 16](see e.g. Fig.3). H e
With all its success and accuracies yet to be improved HRM is only one of the approachesin describing hard photodisintegration reactions. Other models such as reduced nuclearamplitude (RNA) formalism[24] and quark-gluon string (QGS) model[25] describe manyfeatures of hard photodisintegration reaction, with QGS being rather successful in describinglower energy data. However RNA and QGS require an absolute normalization.Recently it was suggested[22, 23] that the break-up of pp pair from He will further ad-vance our understanding of the dynamics of hard photodisintegration of two-nucleon systemsand allow further discrimination between above mentioned models.Within HRM the typical diagram describing two-proton break-up is shown in Fig.4.HRM calculation similar to that of deuteron break-up reaction yields: dσdtd p n = (cid:18) (cid:19) π αS − M He ( 2 c c ) dσ pp dt ( s pp , t n ) S E n , (4)4 pp A r p Bn p p ppn k k p pp Figure 4: Typical diagram for hard break-up of pp pair from He .where S = P λ = − λ ,λ = − (cid:12)(cid:12)(cid:12)(cid:12)R ψ He ( λ , λ , λ ) m d p ⊥ (2 π ) (cid:12)(cid:12)(cid:12)(cid:12) and c = | φ , || φ | with φ i being pp helicityamplitudes. Since the cross section of pp scattering enters in Eq.(4) one of the interest-ing predictions of HRM is the possibility of observation of energy oscillations at 90 CMscattering similar to one observed in elastic pp scattering. Another interesting feature oftwo-proton break-up reactions is the fact that at lower energies this reaction is three-step[26]rather than two-step process. HRM in fact predicts that with an increase of energy dueto the onset of quark-interchange (rather than meson-exchange) mechanism the two-bodyprocesses will dominate the cross section. The pioneering experiment of high energy pp break-up reaction[27] was recently performed at Jefferson Lab which may shed new light onmany issues of hard rescattering processes. In addition to the cross section measurements, polarization observables may provide a newinsight into the dynamics of hard photodisintegration. Original motivation for polarizationmeasurements in high energy photodisintegration reaction was the expectation that the onsetof the pQCD regime in the reaction dynamics will be accompanied by an observation of thehelicity conservation in polarized reaction. Both energy and angular distributions of severalpolarization observables have been measured at JLab[28, 29]. Although the energy rangecovered was rather restricted, it provided an interesting insight into the structure of HRM.One of the unique features of HRM is that the struck quark carries the helicity ofincoming photon. As a result one of the final nucleons will carry the bulk of the polarizationof incident photon (see e.g. Eq.(2)). Thus in HRM photon plays as a helicity selector for thefinal nucleons. This yields a prediction for large asymmetry[27] ( C z ′ ) for the longitudinalpolarization of outgoing nucleons. In Ref.[21] we predicted a sizable asymmetry for C z ′ even though the existing data[28] (with rather large errors) at that time were indicating onvanishing values of C z ′ . However, recent data[29], appears to confirm HRM prediction forlarge values of C z ′ (see Fig.5). It will be interesting also to check the other HRM predictionthat C z ′ will continue to approach to unity with an increase of photon energy at 90 CMscattering.It is very interesting that above described helicity selection mechanism of HRM predicts( an opposite ) vanishing value of C z ′ for two-body break up of proton pair from He . Thisfollows from the fact that the dominant part of the amplitude which represents two final5 Θ cm , deg C z / Figure 5: Angular dependence of C z ′ for E γ = 1 . He target. No such suppressionexists for pn break up reactions. There is an accumulating evidence that hard rescattering mechanism explains the underlyingdynamics of high energy and large CM angle break-up of a nucleon pair from D and He targets. One of the important features of HRM is that its prediction of unpolarized crosssection at 90 center of mass photodisintegration of deuteron is parameter free and no furtheradjustments are required.HRM predicts that energy dependence of two-proton break-up reaction should resemblethat of hard elastic pp cross section.Another feature of HRM, observed recently, is the prediction of large longitudinal asym-metries due to helicity selection mechanism characteristic to hard rescattering model.If HRM will prove to be a true mechanism of hard photodisintegration reaction involvingtwo nucleons, it will advance also our understanding of the dynamics of NN interaction atshort distances.A new venue for advancing our understanding of the dynamics of hard break-up reactionscould be an extension of these studies to the kinematics in which two excited baryonic states(like ∆-isobars) are produced at large center of mass angles of γ – N N system.
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