Decay of Polarized Delta
G. Ramachandran, Venkataraya, M. S. Vidya, J. Balasubramanyam, G. Padmanabha
aa r X i v : . [ nu c l - t h ] J a n Decay of polarized ∆ G Ramachandran , Venkataraya , , M S Vidya ,J Balasubramanyam , and G Padmanabha , , GVK Academy, Jayanagar, Bangalore - 560082, India Vijaya College, Bangalore - 560 011, India K. S. Institute of Technology, Bangalore - 560 062 , India Department of Physics, Bangalore University, Bangalore - 560 056, India and SBM Jain College, V V Puram, Bangalore - 560004, India
The resonance ∆(1232) with spin-parity
32 + , which contributes dominantly tothe reactions like γN → πN and N N → N N π at intermediate energies, may beexpected to be produced in characteristically different polarized spin states. As suchan analysis of the decay of polarized delta is presented, which may be utilized toprobe empirically the production mechanism. It is shown that measurements of theangular distributions of the pion and the polarization of the nucleon arising out of ~ ∆ decay can determine empirically the Fano statistical tensors characterizing the ~ ∆. PACS numbers: 25.20.Ljs,24.70.+s,14.20.Gk ,13.60.Le,13.30.-a
Meson production in
N N collisions [1] as well as photo- and electro-production of mesons[2] have attracted renewed interest with the technological advances [1, 2, 3] made at IUCFIndiana in USA, the development of COSY at Julich in Germany and the advent of newgeneration of electron accelerators at J-Lab, MIT, BNL in USA, ELSA at Bonn, MAMIat Mainz in Germany, ESRF at Grenoble in France and Spring8 at Osaka in Japan. Theexcitement began in the early 1990’s, when the total cross section measurements [4] for pp → ppπ were found to exceed the then available theoretical predictions [5], by morethan a factor of 5. Moreover, a large momentum transfer is involved when an additionalparticle is produced in the final state. This implies that the features of the N N interactionis probed at very short distances, estimated [6] to be of the order of 0 .
53 fm, 0 .
21 fm and0 .
18 fm for the production of π , ω and φ mesons respectively. While ω and φ productioninvolve only excited nucleon states, pion production involves ∆ which is a well isolatedresonance. Moreover, experimental studies on pion production in pp collisions have reacheda high degree of sophistication [7, 8], where both the protons are polarized initially andthe three body final state is completely identified kinematically. The Julich meson exchangemodel [9], which yielded theoretical predictions closer to the experimental data than mostother models, has been more successful in the case of charged pion production [7] thanwith the production of neutral pions [8]. A more recent analysis [10] of the ~p~p → ppπ measurements [8], following a model independent irreducible tensor approach [11], showedthat the Julich model deviates quite significantly for the P → P p and to a lesser extent forthe F → P p transitions. These calculations [10] have been carried out with and withouttaking the ∆ contributions into account and this exercise emphasized the important roleof ∆ in the model calculations. It has been shown [12] that N N → N ∆ reaction matrixcontains as many as 16 amplitudes out of which 10 are second rank tensors. Ray [13] hasdrawn attention to their importance based on a partial wave expansion model where hefound that “ the total and differential cross-section reduced by about one half, the structurein the analyzing powers increased dramatically, the predictions of D NN became much toonegative, while that for D LL became much too positive and the spin correlation predictionswere much too small when all ten of the rank 2 tensor amplitudes were set to zero, while theremaining six amplitudes were unchanged ”. The dominance of ∆ in photo and electro-pionproduction has been known for a long time. Very recently, a theoretical formalism [14] forphoto and electroproduction of mesons with arbitrary spin-parity s π has been outlined, whichspecializes to pion production for s π = 0 − . The excitation of a nucleon into ∆ through photo-absorption, involves electric quadrupole ( E
2) and magnetic dipole ( M
1) form factors, whileelectro-production involves a Coulomb form factor ( C
2) in addition. The electro-excitationof ∆ provides a test [15] for perturbative QCD, which appears to fail when confronted withexperiments [16]. The study of these form factors as well as the quadrupole deformation of∆ has excited considerable attention [17]. The importance of ∆ has also been highlightedin more recent studies [18].One can naturally expect ∆, whether it is produced through electromagnetic excitationor hadronic excitation, to be polarized. Characterizing the ~ ∆ produced by the Fano [19]statistical tensors t kq of rank k = 1 , , q = − k, − k + 1 , ....., k , we discuss here theextent of information that can be obtained on the t kq through measurements of the angulardistribution of the pion and that of the nucleon polarization in the final state.For this purpose, we express the reaction matrix M for the decay of ∆ in the form M = f ( S ( , ) · Y ( ˆq )) , (1)where q denotes the meson momentum in the ∆ rest frame and ˆq = q / | q | . The transitionspin operators S λm ( s f , s i ) of rank λ , connecting the initial and final channel spins s i and s f respectively, are defined following [20], the multiplicative factor f defines the strength ofthe transition and Y l,m ( ˆq ) denote spherical harmonics. The state of polarization of ∆ is, ingeneral, represented by the spin density matrix ρ ∆ = T r ( ρ ∆ )4 X k =0 ( τ k · t k ) , (2)in terms of Fano statistical tensors t kq = T r ( τ kq ρ ∆ ) T rρ ∆ , (3)where τ kq denote irreducible tensor operators of rank k in the ∆ spin space of dimension2 j + 1 = 4. The hermitian conjugates ( τ kq ) † are related to ( τ kq ) through ( τ kq ) † = ( − q τ k − q and are so normalized as to satisfy T r [ τ kq ( τ k ′ q ′ ) † ] = (2 j + 1) δ kk ′ δ qq ′ , (4)with τ = 1, so that t = 1. The above normalization is different from that used by Fanooriginally, but consistent with that chosen [21] for spin j = 1. It may be noted that τ q for spin-3 / J = J z ; J ± = ∓ ( J x ± iJ y ) √ of the spinoperator J of ∆ through τ q = r J q , (5)whereas of τ q = q J q in the case of spin j = 1.Our purpose is to determine empirically the t kq for k = 1 , , ~ ∆ through an experimentalstudy of its decay productsThe angular distribution of the pion in the case of ~ ∆ decay is given by I ( ˆq ) = T r ( M ρ ∆ M † ) , (6)where M † denotes the hermitian conjugate of M . Noting that τ kq ≡ S kq ( , ) and makinguse of the known properties [20] of the spin operators, we obtain I ( ˆq ) = X k =0 , k X q = − k ( − q I k − q Y kq ( ˆq ) , (7)in terms of I kq = 9 | f | √ π W ( k C (1 k
1; 000) t kq , (8)which may be measured experimentally, since Y kq ( ˆq ) are orthonormal to each other. Notethat t = 1, yields | f | empirically from an experimental measurement of I . This informa-tion may then be used to determine t q from experimentally measured I q . We next observethat the spin density matrix ρ N of the nucleon in the final state is given by ρ N = M ρ ∆ M † , (9)which may be expressed in the standard form ρ N = T r ( ρ N )2 [1 + σ · P ] , (10)in terms of the Pauli spin matrices σ of the nucleon and nucleon polarization P = T r ( σ ρ N ) T r ( ρ N ) , (11)where the denominator is already known from eq.(6) as I ( ˆq ). Expressing ( σ . P ) in terms ofspherical components through ( σ · P ) = P ν ( − ν σ − ν P ν and noting further that σ ν ≡ S ν ( , ) , (12)we obtain P ν ( ˆq ) = 9 r π | f | [ I ( ˆq )] − X k =1 k X q = − k t kq X l =0 , G ( k, l ) × C ( kl qmν ) Y lm (ˆ q ) , (13)where the geometrical factors are G ( k, l ) = X Λ ( − ( k − Λ) [ k ] [Λ] W ( k Λ) W ( Λ; × W ( k l ) C (11 l ; 000) (14)We note that I ( ˆq ) and | f | are already known, so that the spherically symmetric termwith l = 0 in eq.(13) yields t q . The l = 2 terms contain all the t kq with k = 1 , ,
3. However,since it has already been shown that t q and t q are determinable empirically from I (ˆ q ) and l = 0 term of eq.(13), experimental study of the angular distribution with l = 2 may beused to determine t q empirically.One can expect the empirical study of ∆ polarization to yield useful information withregard to the formation of ∆ in different scenarios. These detailed case by case studies willbe taken up elsewhere. Apart from this, we may also draw attention to some interestingaspects of ∆ polarization. While t q determine the magnitude as well as the direction of thevector polarization, which is an axial vector, it is interesting to point out that two otherindependent axes are associated [22] with the second rank polarization t q , which definealso the Principal Axes of Alignment. Hence the measurements of the t q determine also thecorresponding Principal Axes of Alignment Frame (PAAF) of ~ ∆. Moreover, three additionalindependent axes are associated [23] with the third rank tensor t q , so that a maximal set ofsix independent axes characterize ~ ∆, apart from the relative strengths of vector, second rankand third rank tensor polarizations. If the ~ ∆ produced is oriented [24], which is the simplestcase of polarization of ∆, all the six independent axes collapse into one. Such a scenario isquite unlikely especially in N N → N N π . We, therefore, advocate the experimental studyof the angular distribution of the pion and that of the nucleon polarization for events, whichcorrespond kinematically to invariant mass of the π − N system being equal to the mass ofthe ∆ resonance. Acknowledgements
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