� In tro duction � Spin correlations for the �� and � � pairs� generated in hea vy ion colli� sions� and resp ectiv e angular correlations at the join t registration of hadron� ic deca ys of t w o h yp erons with space parit y nonconserv ation giv e imp ortan t information ab out the c haracter and mec hanism of m ultipl e pro cesses� The � adv an tage of the �� and � � systems o v er other ones is due to the fact that � � the P �o dd deca ys � � p � � and � � p � � � serv e as e�ectiv e analyzers � � of spin state of the � and � particles� In connection with this� spin corre� � lations in the �� and � � systems can b e rather easily distinguished and studied exp erimen tall y b y the metho d of �momen ts� o v er the bac kground of a large amoun t of pro duced secondary particles� This fact is esp ecially meaningful for the in v estigations of m ultipl e generation at mo dern and fu� ture ion colliders lik e RHIC� LHC� NICA� since the p olarization parameters � � esp ecially for the � � pair � are v ery sensitiv e to the scenario of pro cess after the act of collision of relativistic hea vy ions � � General structure of the spin densit y matrix of the � pairs �� and � � � The spin densit y matrix of the �� and � � pairs� just as the spin densit y matrix of t w o spin���� particles in general� can b e presen ted in the follo wing form �������� � � � � � �� � �� ��� ��� ��� ��� ��� ��� � � � � I � I � � � P � � I � I � � � P �� � � � � � � � X X ��� ��� � T � � � � � �� ��� ik i k i �� k �� �� � �� in doing so� tr � � � �� �� � �� � Here I is the t w o�ro w unit matrix� � � � � � � � � � � � is the v ector P auli � x y z op erator � x� y � z � � � � � � �� P and P are the p olarization v ectors of �rst � � ��� ��� ��� ��� and second particle � � h � i � � h � i �� T � h � � � � � i are P � P � � � ik i k the correlation tensor comp onen ts � In the general case T � � P P � The ik � i � k tensor with comp onen ts C � T � P P describ es the spin correlations ik ik � i � k of t w o particles � The resp ectiv e one�particle densit y matrices are as follo ws� � � ��� � ��� ��� ��� � ��� ��� � � � � I � � � � � � � � I � � � � ��� P � P � � � � � �
The �trace� of the correlation tensor is T � T � T � T � xx y y z z ��� ��� ��� ��� � h � � � i � The eigen v alues of the op erator � � � equal � � � � � � � t for three triplet states �total spin S � �� and � � � � for the singlet state s �total spin S � �� � Let us in tro duce the op erators of pro jection on to the triplet states and on to the singlet state� ��� ��� ��� ��� � � � � � � � � � � � � � � � � P � � P � � ��� t s � � The follo wing matrix equalities are satis�ed� � � � � � � � � � � P � P � P � P � P P � P P � � � t s t s s t t s F or all the purely triplet states the �trace� of the correlation tensor T � �� whereas for the purely singlet state T � � � � No w let us in tro duce the op erator of p erm utation of spin pro jections� ha ving the form� � � � � �� � �� ��� ��� ��� ��� � � P � � I � I � � � � � � ��� � Then w e obtain� � � � � � � � � � � �� � �� �� � �� �� � �� �� � �� P P � P P � � P � P P � P P � � P � ��� t t t s s s � �� � �� The eigen v alues of P equal �� for the three triplet states and � � for the singlet state � The t w o�particle spin densit y matrix ��� ma y b e decomp osed in to the triplet� singlet and the �mixed� singlet�trip let parts� �� � �� � � � � � � � � � � � � � � � ��� t s ts st in doing so� � � � � � � � � �� � �� �� � �� �� � �� �� � �� � � � P � � P � � � � P � � P � � � � P � � P � � � � P � � P � t t t s s s ts t s st s t ��� The relativ e fraction of the triplet states amoun ts to� � � T � � � �� � �� �� � �� W � tr � � � tr � P � � P � � tr � P � � � � � ��� t t t t t � and the relativ e fraction of the singlet state amoun ts to� �
� � T � � � �� � �� �� � �� W � tr � � � tr � P � � P � � tr � P � � � � � ��� s s s s s � in doing so� W � W � �� T � W � � W � � � � W � ���� t s t s s � � Due to the orthogonalit y of the pro jection matrices P and P � the fol� t s lo wing relation holds� � � �� � �� tr � � � tr � � � tr � � � P P � � � � ts st s t There are also some matrix equalities con taining the op erator of p erm u� � �� � �� tation of spin pro jections P � � � � � �� � �� �� � �� �� � �� �� � �� P � � P � � � � � P � � P � � � � � t t s s � � � � �� � �� �� � �� �� � �� �� � �� P � � P � � � � � P � � P � � � � � ���� ts ts st st It should b e noted that the matrices � � and � � incorp orate only symmetric t s com binations of spin op erators of t w o particles� whereas the matrices � � and ts � � include only an tisymmetric com binations of spin op erators � st The explicit form ulas for the triplet � � � �� singlet � � � � and �mixed� t s � � � � � � � comp onen ts of the spin densit y matrix are as follo ws� ts st h � � � T P � P � � � � ��� � � ��� ��� ��� ��� ��� � � � � � I � I � � � � I � I � � � � t � � � � � i � � X X T � T � � T ��� ��� ik k i � � � � � � � � � ���� ik i k � � i �� k �� � � � T � � ��� ��� ��� ��� � � � � � � I � I � � � � �� ���� s � � h � P � P � � � � ��� ��� ��� ��� � � � � � � � � � � � � I � I � � � � � � � ts st � � i � � X X T � T ik k i ��� ��� � � � � � � � � i � � k � ���� i k � i �� k �� �
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