A method of estimating the physical bounds for the linear combinations of components of statistical tensors is presented. Numerical values of the hounds for tensor combinations appearing in the additive quark model predictions for resonance production in two-body processes are calculated

Kotański, Andrzej

Zalewski, K.

Średniawa, B.

Acta Physica Polonica B

English

Jagiellonian Univeristy Repository

Vol. B2 (1971) ACTA PHYSICA POLONICA Fasc. 6BOUNDS FOR THE LINEAR COMBINATIONS OF STATISTICALTENSORSBy A. Kotański, B. ŚredniawaInstitute of Physics, Jagellonian University, Cracow** Address: Instytut Fizyki, Uniwersytet Jagielloński, Kraków 16, Reymonta 4, Poland. (789)And K. ZalewskiInstitute of Nuclear Physics, Cracow(Received April 6, 1971)A method of estimating the physical bounds for the linear combinations of components of statistical tensors is presented. Numerical values of the bounds for tensor combinations appear- ing in the additive quark model predictions for resonance production in two-body processes are calculated.Much interest has been recently paid to the angular decay distributions of the resonances produced in high-energy processes. Joint decay distributions have been studied experiment- ally in several processes and some theoretical models have been proposed and checked. A typical example is the additive quark model (cf. e.g. Ref. [1]), which predicts relations be- tween the statistical tensors [1,2]. When checking experimentally such relations it is interesting to know the kinematical bounds for the linear combinations of statistical tensors.The purpose of this notę is to present a convenient method of estimating these bounds and to list the numerical values of the bounds for some frequently used tensor combinations.The method used here is essentially that of Refs [3] and [4]. A double statistical tensor is defined as the following combination of the density matrix elements:× C(s1, - m1∙, s1, m1∖J1,Mι)C(sz, - m2∙, ⅛ m'∣J2, M2)ρ^<.It may be regarded as the average value of an operator (⅛⅛1(1)(2)790TABLE ICalculated kinematical bounds for linear combinations of statistical tensors. According to the additive quark model each combination should be equal to the value quoted in column 5. Column 6 contains references to original papers where the relatioπs have been derivedSpins Reaction Combination of tensors Calculated bounds AQM prediction Ref.1, 3̂2^ PB → PB* (-0.8165, 0.6124) 0 [2,6]γB → PB* T22 _ 1 T2000 2 20 0.3750 0 [2,6]T22 — ------ γιQ202 [/2 102 0.4330 0 [2,6]T22 I 1 T02 0.4082 1 Γ9 61o° ∣∕2 00 2J/6 Lz> OJPB → PB* rs-∕⅛ 1.1619 0 [5]γS- F⅛ 0.5387 0 [2]T22 _ T,2220 02 0.2887 0 [2]η2 + ¾2+ τiι2 0.4082 12|/6 [2]9 2’32BB → B* 7,22 1 rΓ2020 2 Ł 20 0.2652 0 [2]7122 1 1 710200 ' 2 ∙t00 0.3750 18 [2]7no^2jnθ2 1.0062 0 [5]T,ll δ 772211 3 1 11 0.5667 0 [5]7^1 1 < 5 T2 2 i 1—1 T 3 1 1—1 0.5667 0 [5]T,20 rΓ22 37,H e>T2 200 *00  5 00 xj2-2 (- 0.7190; 1.0220) 0 [5]T22 t 'Γ22t T2 2j22^ j00^t^∙z2-2 0.5000 14 [2]1, 32 γB → PB* η√Hz⅛ 0.5387 0 [6]T22 « T2220 ' 02 0.2887 0 [6]'Γ22 T22 T2 27 00 7 22 7 2“2 0.4082 J/6 [6]rΓ22 1 J,2020 * 20 0.5000 0 [6]w⅛ 0.4339 0 [6]1∕2TJ2+¾° (-0.8165, 0.6124) ~ 2 |/ó [6]γ^÷2c+6too2 (-1.1547, 0.8660) 0 [6]791TABLE I (continued)Spins Reaction Combination of tensors Calculated bounds AQM prediction Ref.1 2 γB → KB* r»o“]/|Re^ 0.4082 0 [6]ro2o2-]/|Re7?o20.4082 0 [6]Γ22 + (7,f-i)*-  6To≡2 0.7071 0 [6]J,22__(J,2 2j* 0.3536 0 [6]with matrix elements equal to the products of Clebsch-Gordan coefficients as seen from Eq. (1). The average value of an operator cannot exceed its extremal eigenvalues. The method of estirnating the bounds for the tensors like (1) proposed in Ref. [3] consists in calculating the eigenvalues of the hermitian part of the operator (P. The values for bounds obtained in this way for single and double tensors are given in Ref. [4].A similar method can be used to evaluate the bounds for the linear combinations of various tensor components(3)The τnatrix elements of are again appropriate products of Clebsch-Gordan coefficientsand the problem is reduced again to finding the minimum and maximum eigenvalues of the Hermitian part of a known matrix.In Table I we present as an exarnple the estimated bounds for the linear combinations of tensors appearing in the predictions of the additive quark model. Each entry contains the spins of the two resonances described by the tensors, the reaction for which the prediction applies, the linear combination of tensor components, the estimated bounds, the additive quark model prediction for this combination, and finally the reference to the paper where the prediction has been derived.The results were obtained using a computer programme. The programme calculated the matrix elements of the Hermitian part of the operator from Eq. (4) and diagonalized the matrix obtained. The numbers in the Table are the bounds for the moduli, for the real parts and for the imaginary parts of the tensor combinations, at the same time.The method is especially effective when the linear combination contains morę than two tensor components. Most of our results for combinations with two terjns may also be obtained by taking the appropriate combination of bounds for tensor components from Ref. [4],(4)where a,b, ... n are known constants. This expression may be represented as the average value of the operator792REFERENCES[1] C. Itzykson, M. Jacob, Nuoυo Cimento, 48A, 909 (1967); J. L. Friar, J. S. Trefił, Nuoυo Cimento, 49A, 642 (1967).[2] A. Białas, K. Zalewski, Nuclear Phys., B6, 465 (1968).[3] S. Fenster, Thesis, UCLA report, May 1965.[4] A. Kotański, B. Sredniawa, K. Zalewski, Nuclear Phys., B23, 541 (1970).[5] A. Białas, A. Kotański, Nuclear Phys., B19, 667 (1970).[6] B. Gorczyca, M. Hayashi, Acta Phys. Polon., 36, 433 (1969).

Bounds for the linear combinations of statistical tensors

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Bounds for the linear combinations of statistical tensors

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