Quark Structure of the Nucleon and Angular Asymmetry of Proton-Neutron Hard Elastic Scattering
aa r X i v : . [ h e p - ph ] J u l Quark Structure of the Nucleon and Angular Asymmetry of Proton-Neutron HardElastic Scattering
Carlos G. Granados and Misak M. Sargsian
Florida International University, Miami, FL 33199 USA (Dated: November 29, 2018)We investigate an asymmetry in the angular distribution of hard elastic proton-neutron scatteringwith respect to 90 center of mass scattering angle. We demonstrate that the magnitude of theangular asymmetry is related to the helicity-isospin symmetry of the quark wave function of thenucleon. Our estimate of the asymmetry within the quark-interchange model of hard scatteringdemonstrates that the quark wave function of a nucleon based on the exact SU(6) symmetry predictsan angular asymmetry opposite to that of experimental observations. On the other hand the quarkwave function based on the diquark picture of the nucleon produces an asymmetry consistent withthe data. Comparison with the data allowed us to extract the relative sign and the magnitude ofthe vector and scalar diquark components of the quark wave function of the nucleon. These twoquantities are essential in constraining QCD models of a nucleon. Overall, our conclusion is that theangular asymmetry of a hard elastic scattering of baryons provides a new venue in probing quark-gluon structure of baryons and should be considered as an important observable in constraining thetheoretical models. For several decades elastic nucleon-nucleon scatteringat high momentum transfer ( − t, − u ≥ M N GeV ) hasbeen one of the important testing grounds for QCD dy-namics of the strong interaction between hadrons. Twomajor observables considered were the energy depen-dence of the elastic cross section and the polarizationproperties of the reaction.Predictions for energy dependence are based on the un-derlying dynamics of the hard scattering of quark com-ponents of the nucleons. One such prediction is basedon the quark-counting rule [1, 2] according to whichthe differential cross section of two-body elastic scatter-ing ( ab → cd ) at high momentum transfer behaves like dσdt ∼ s − ( n a + n b + n c + n d ) , where n i represents the numberof constituents in particle i (i=a,b,c,d).For elastic N N scattering, the quark-counting rule pre-dicts s − NN scaling which agrees reasonably well with ex-perimental measurements (see e.g. Refs.[3, 4, 5, 6]). Inaddition to energy dependence, the comparison [7] of thecross sections of hard exclusive scattering of hadrons con-taining quarks with the same flavor with the scattering ofhadrons that share no common flavor of quarks demon-strated that the quark-interchange represents the domi-nant mechanism of hard elastic scattering for up to ISRenergies (see discussion in [8]).For polarization observables, the major prediction ofthe QCD dynamics of hard elastic scattering is the con-servation of helicities of interacting hadrons. The latterprediction is based on the fact that the gluon exchange inmassless quark limit conserves the helicity of interactingquarks.Quark counting rule and helicity conservation howeverdo not describe completely the features of hard scatteringdata. The energy dependence of pp elastic cross sectionscaled by s NN exhibits an oscillatory behavior which in-dicates the existence of other possibly nonperturbativemechanisms for the scattering[9, 10]. These expectationsare reinforced also by the observed large asymmetry, A nn at some hard scattering kinematics[11] which indicates ananomalously large contribution from double helicity flipprocesses. These observed discrepancies however do notrepresent the dominant features of the data and overallone can conclude that the bulk of the hard elastic N N scattering amplitude is defined by the exchange mecha-nism of valence quarks which interact through the hardgluon exchange (see e.g. Refs.[8, 12]). Quark-interchangemechanism also reasonably well describes the 90 c.m.hard break-up of two nucleons from the deuteron[13, 14].However, the energy dependence of a hard scatteringcross section, except for the verification of the domi-nance of the minimal-Fock component of the quark wavefunction of nucleon, provides rather limited informationabout the symmetry properties of the valence quark com-ponent of the nucleon wave function.In this work we demonstrate that an observable suchas the asymmetry of a hard elastic proton-neutron scat-tering with respect to 90 c.m. scattering may provide anew insight into the helicity-flavor symmetry of the quarkwave function of the nucleon. Namely we consider A ( θ ) = σ ( θ ) − σ ( π − θ ) σ ( θ ) + σ ( π − θ ) , (1)where σ ( θ ) - is the differential cross section of the elastic pn scattering. We will discuss this asymmetry in the hardkinematic regime in which the energy dependence of thecross section is ∼ s − . Our working assumption is thedominance of the quark-interchange mechanism (QIM)in the N N elastic scattering at these kinematics.Within QIM the characteristic scattering diagram canbe represented as in Fig.1. Here one assumes a factor-ization of the soft part of the reaction in the form of theinitial and final state wave functions of nucleons and thehard part which is characterized by QIM scattering thatproceeds with five hard gluon exchanges which generateenergy dependence in accordance to the quark countingrule. In order to attempt to calculate the absolute cross b dca
FIG. 1: Typical diagram for quark-interchange mechanism of NN → NN scattering. section of the reaction one needs to sum hundreds of dia-grams similar to one of Fig.1. However for the purpose ofestimation of the asymmetry in Eq.(1) the important ob-servation is that the hard scattering kernel is flavor-blindand conserves the helicity. As a result one expects thatangular asymmetry will be generated mainly through theunderlying spin-flavor symmetry of the quark wave func-tions of the interacting nucleons.The amplitude of the hard elastic a + b → c + d scat-tering of Fig.1, within quark-interchange approximation,can be presented as follows: h cd | T | ab i = X α,β,γ h ψ † c | α ′ , β ′ , γ ′ ih ψ † d | α ′ , β ′ , γ ′ i×h α ′ , β ′ , γ ′ , α ′ β ′ γ ′ | H | α , β , γ , α β γ i · h α , β , γ | ψ a ih α , β , γ | ψ b i , (2)where ( α i , α ′ i ), ( β i , β ′ i ) and ( γ i , γ ′ i ) describe the spin-flavor quark states before and after the hard scattering, H , and C jα,β,γ ≡ h α, β, γ | ψ j i (3)describes the probability amplitude of finding the α, β, γ helicity-flavor combination of three valence quarks in the nucleon j [12].To be able to calculate C jα,β,γ factors one representsthe nucleon wave function through the helicity-flavor ba-sis of the valence quarks. We use a rather general formseparating the wave function into two parts characterizedby two (e.g. second and third) quarks being in spin zero- isosinglet and spin one - isotriplet states as follows: ψ i N ,h N = 1 √ n Φ , ( k , k , k )( χ (23)0 , χ (1) ,h N ) · ( τ (23)0 , τ (1) ,i N ) + Φ , ( k , k , k ) × X i = − X h = − h , h ; 12 , h N − h | , h N ih , i ; 12 , i N − i | , i N i ( χ (23)1 ,h χ (1) ,h N − h ) · ( τ (23)1 ,i τ (1) ,i N − i ) , (4)where j N and h N are the isospin component and thehelicity of the nucleon. Here k i ’s are the light cone mo-menta of quarks which should be understood as ( x i , k i ⊥ )where x i is a light cone momentum fraction of the nu-cleon carried by the i -quark. We define χ j,h and τ I,i as helicity and isospin wave functions, where j is thespin, h is the helicity, I is the isospin and i its thirdcomponent. The Clebsch-Gordan coefficients are de-fined as h j , m ; j , m | j, m i . Here, Φ I,J representsthe momentum dependent part of the wave function for( I = 0 , J = 0) and ( I = 1 , J = 1) two-quark spectatorstates respectively. Since the asymmetry in Eq.(1) doesnot depend on the absolute normalization of the crosssection, a more relevant quantity for us will be the rel-ative strength of these two momentum dependent wavefunctions. For our discussion we introduce a parameter, ρ : ρ = h Φ , ih Φ , i (5)which characterizes an average relative magnitude of thewave function components corresponding to ( I = 0 , J =0) and ( I = 1 , J = 1) quantum numbers of two-quark“spectator” states. Note that the two extreme values of ρ define two well know approximations: ρ = 1 correspondsto the exact SU(6) symmetric picture of the nucleon wavefunction and ρ = 0 will correspond to the contributionof only good-scalar diquark configuration in the nucleonwave function (see e.g. Ref.[15, 16, 17, 18] where thiscomponent is referred as a scalar or good diquark con-figuration ([ qq ]) as opposed to a vector or bad diquarkconfiguration denoted by ( qq )). In further discussions wewill keep ρ as a free parameter.To calculate the scattering amplitude of Eq.(2) we as-sume a conservation of the helicities of quarks participat-ing in the hard scattering. This allows us to approximatethe hard scattering part of the amplitude, H , in the fol-lowing form: H ≈ δ α α ′ δ α α ′ δ β ,β ′ δ γ ,γ ′ δ β ,β ′ δ γ ,γ ′ f ( θ ) s . (6)Inserting this expression into Eq.(2) for the QIM ampli-tude one obtains[12]: h cd | T | ab i = T r ( M ac M bd ) (7)with: M i,jα,α ′ = C iα,βγ C jα ′ ,βγ + C iβα,β C jβα ′ ,β + C iβγα C jβγα ′ , (8)where we sum over the all possible values of β and γ .Furthermore, we separate the energy dependence fromthe scattering amplitude as follows: h cd | T | ab i = h h c , h d | T ( θ ) | h a , h b i s (9)and define five independent angular parts of the helicityamplitudes as: φ = h ++ | T ( θ ) | ++ i ; φ = h−− | T ( θ ) | ++ i ; φ = h + − | T ( θ ) | + −i ; φ = −h− + | T ( θ ) | + −i ; φ = h− + | T ( θ ) | ++ i . (10)Here the “-” sign in the definition of φ follows from theJacob-Wick helicity convention[19] according to which a(-1) phase is introduced if two quarks that scatter to π − θ cm angle have opposite helicity (see also Ref.[12]).Using Eqs.(7,8) for the non-vanishing helicity ampli-tudes of Eq.(10) one obtains:for pp → pp : φ = (3 + y ) F ( θ ) + (3 + y ) F ( π − θ ) φ = (2 − y ) F ( θ ) + (1 + 2 y ) F ( π − θ ) φ = − (1 + 2 y ) F ( θ ) − (2 − y ) F ( π − θ ) (11)and for pn → pn : φ = (2 − y ) F ( θ ) + (1 + 2 y ) F ( π − θ ) φ = (2 + y ) F ( θ ) + (1 + 4 y ) F ( π − θ ) φ = 2 yF ( θ ) + 2 yF ( π − θ ) (12)with φ = φ = 0 due to helicity conservation. Here: y = x ( x + 1) with x = 2 ρ ρ ) (13)and F ( θ ) is the angular function. Note that the ρ = 1case reproduces the SU(6) result of Refs.[12] and [8].The results of Eqs.(11) and (12) could be obtained alsothrough the formalism of the H-spin introduced in Ref.[8]. -1-0.500.510 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1cos( q c.m. ) A P LAB =6GeV/cP
LAB =7GeV/cP
LAB =8GeV/cP
LAB =9GeV/cP
LAB =10GeV/cP
LAB =11GeV/cP
LAB =12GeV/c
FIG. 2: Asymmetry of pn elastic cross section. Solid dottedline - SU(6), with ρ = 1, dashed line diquark-model with ρ = 0, solid line - fit with ρ = − . In this case the helicity amplitudes will be expressedthrough the average number of quarks to be found in agiven helicity-spin state. These numbers will be directlydefined through the wave function of Eq.(4).Introducing the symmetric and antisymmetric parts ofthe angular function F as follows: s ( θ ) = F ( θ ) + F ( π − θ )2 ; a ( θ ) = F ( θ ) − F ( π − θ )2 (14)and using Eq.(12) for the asymmetry as it is defined inEq.(1) one obtains: A ( θ ) = 6 a ( θ ) s ( θ )(1 − y − y ) a ( θ ) (1 − y ) + 3 s ( θ ) (3 + 6 y + 7 y ) . (15)One can make a rather general observation fromEq.(15), that for the SU(6) model, ( ρ = 1, y = ) andfor any positive function, a ( θ ) at θ ≤ π , the angularasymmetry has a negative sign opposite to the experi-mental asymmetry (Fig.2). Note that one expects a pos-itive a ( θ ) at θ ≤ π from general grounds based on theexpectation that in the hard scattering regime the num-ber of t -channel quark scatterings dominates the numberof u -channel quark scatterings in the forward direction.As it follows from Eq.(15), positive asymmetry canbe achieved only for 1 − y − y >
0, which accord-ing to Eq.(13) imposes the following restrictions on ρ : ρ < .
49 or ρ > . pn scattering calculatedwith SU(6) ( ρ = 1) and pure scalar-diquark ( ρ = 0)models are compared with the data. In these estimateswe use F ( θ ) = C · sin − ( θ )(1 − cos ( θ )) − dependence ofthe angular function[20] which is consistent with the pic-ture of hard collinear QIM scattering of valence quarkswith five gluon exchanges and reasonably well reproducesthe main characteristics of the angular dependencies ofboth pp and pn elastic scatterings. Note that using aform of the angular function based on nucleon form-factorarguments[8, 12], F ≈ (1 − cos ( θ )) − will result in thesame angular asymmetry.The comparisons show that the nucleon wave func-tion (4) with a good-scalar diquark component ( ρ = 0)produces the right sign for the angular asymmetry. Onthe other hand even large errors of the data do not pre-clude to conclude that the exact SU(6) symmetry ( ρ = 1)of the quark wave function of nucleon is in qualitativedisagreement with the experimental asymmetry.Using the above defined angular function F ( θ ) we fit-ted A in Eq.(15) to the data at − t, − u ≥ varying ρ as a free parameter. We used the MaximalLikelihood method of fitting excluding those data pointsfrom the data set whose errors are too large for meaning-ful identification of the asymmetry. The best fit is foundfor ρ ≈ − . ± . . (16)The nonzero magnitude of ρ indicates the small but finiterelative strength of a bad/vector diquark configurationin the nuclear wave function as compared to the scalardiquark component. It is intriguing that the obtainedmagnitude of ρ is consistent with the 10% probability of“bad” diquark configuration discussed in Ref.[17].Another interesting property of Eq.(16) is the negativesign of the parameter ρ . Within qualitative quantum-mechanical picture, thenegative sign of ρ may indicate for example the existenceof a repulsion in the quark-(vector- diquark) channel asopposed to the attraction in the quark - (scalar-diquark)channel. It is rather surprising that both the magnitudeand sign agree with the result of the phenomenological in-teraction derived in the one-gluon exchange quark modeldiscussed in Ref.[16].In conclusion, we demonstrated that the angular asym-metry of hard elastic pn scattering can be used to probethe symmetry structure of the valence quark wave func-tion of the nucleon. We demonstrated that the exactSU(6) symmetry does not reproduce the experimental an-gular asymmetry of hard elastic pn scattering. Nucleonwave function consistent with the diquark structure givesa right asymmetry. The fit to the data indicates 10%probability for the existence of bad/vector diquarks inthe wave function of nucleons. It also shows that the vec-tor and scalar qq components of the wave function maybe in the opposite phase. This will indicate on differentdynamics of q − [ qq ] and q − ( qq ) interactions.The relative magnitude and the sign of the vector ( qq )and scalar [ qq ] components can be used to constrain thedifferent QCD predictions which require the existence ofdiquark components in the nucleon wave function. Thesequantities in principle can be checked in Lattice calcula-tions. The angular asymmetry studies can be extendedalso to include the scattering of other baryons such as∆-isobars (which may have a larger fraction of vector di-quark component) as well as strange baryons which willallow us to study the relative strength of ( qq ) and [ qq ]configurations involving strange quarks.This work is supported by U.S. Department of Energygrant under contract DE-FG02-01ER41172. [1] S.J. Brodsky and G.R. Farrar, Phys. Rev. Lett. , 1153(1973); Phys. Rev. D11 , 1309 (1975);[2] V. Matveev, R.M. Muradyan and A.N. Tavkhelidze, Lett.Nuovo , 719 (1973).[3] J. V. Allaby et al. , Phys. Lett. B , 67 (1968).[4] C. W. Akerlof et al. , Phys. Rev. , 1138 (1967).[5] M. L. Perl et al. , Phys. Rev. D , 1857 (1970).[6] J. L. Stone et al. , Nucl. Phys. B , 1 (1978).[7] C. White et al. , Phys. Rev. D , 58 (1994).[8] S. J. Brodsky, C. E. Carlson and H. Lipkin, Phys. Rev.D , 2278 (1979).[9] B. Pire and J. P. Ralston, Phys. Lett. B , 233 (1982);Phys. Rev. Lett. , 1823 (1988).[10] S. J. Brodsky and G. F. de Teramoud, Phys. Rev. Lett. , 1924 (1988).[11] D. G. Crabb et al. , Phys. Rev. Lett. , 1257 (1978).[12] G. R. Farrar et al. , Phys. Rev. D , 202 (1979).[13] L. L. Frankfurt et al. , Phys. Rev. Lett. , 3045 (2000).[14] M. M. Sargsian, Phys. Lett. B , 41 (2004).[15] M. Anselmino et al. Rev. Mod. Phys. , 1199 (1993).[16] R. L. Jaffe, Phys. Rept. , 1 (2005).[17] A. Selem and F. Wilczek, arXiv:hep-ph/0602128.[18] A. Bacchetta, F. Conti, and M. Radici, Phys. Rev. D78,074010 (2008)[19] M. Jacob and G.C. Wick, Annals Phys. , 404 (1959).[20] G. P. Ramsey and D. W. Sivers, Phys. Rev. D45