Mitra, A N

Sharma, Anju

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Indian J. Phys. 70A (3), 383-385 (1996)I J P  A-  an international journalA new quantum number for qqq baryonsA. N. M itra* and Anju SharmaNat Inst of Adv Studies, I I Sc Campus, Bangalore 12, IndiaAbstract A complex harmonic oscillator basis is proposed for a three-body systemobeying S3-symmetry, so as to generate an extraquantum number (Na), over and above the (standard) total quantum number (N), for a more quantitative S3-classification of the various qqq states than is possible m the usual (real) representation Further, certain bilinears in their complex forms with definite S3-symmetry properties that can be constructed out of the linear h.o. operators (a, a*, a , a t ), satisfy several distinct SO (2, 1) algebras with spectra bounded from below. The 3-body jpave function is written down in the complex basisKeywords Three-body system. Si-symmetryPACS Nos 11 30.Fs, 02.20 -aIntroductionIt is very well known that the quantum state of a three-body system in the real basis is specified by I NILS ; X >. Here X characterizes the S3-sym m etry status of the system in terms of ‘symmetric* (s), ‘antisym m etric’ (a) and ‘m ixed-sym m etric’ (m), all of which are expressible in terms o f two independent (Jacobi) variables (£*, rjj) forming a [2, 1 ] representation o f S3-sym m etry. However, a disadvanntage of this real representation is that the above S 3-sym m etry  charactrization (X) is not quantitatively expressed in terms of some additional quantum  numbers. This shortcoming is more acute in the case o f relativistic three-body problem  wherein a large number o f S 3-sym m etry breaking terms are present in the interaction hamiltonian than for the corresponding non-relativistic system. This can be taken care o f by the use o f a more promising approach, viz., the complex basis defined through< 2  Zj = £1 + i rii ; V2 Z* = Z,i -  i T)i. (1)A new quantum  num ber can be generated in a simple way, without going into any particular representation, thtough the quantity tan v = 2% r\f ( £ 2  -  t l2) and the related operator -  i v  whose eigennfunctions are exp (ivX) with eigenvalues 7J2 where X = 3 n - l ,  3n, 3n + 1 (n=o, 1, 2. .) [1].Address after 1995 : 244-Tagore Park, Delhi 110009.© 19% IACS384 A  N  M it r a  a n d  A n ju  S h a rm aThe Harmonic representation, on the other hand, offers a m ore specific information since the dynam ics is controlled by the two number operatiorsN«" V * . N „ =  a (2)where a, a+ are normalized ladder operators in £, Tl indices deffined in the standard manner. In this (£ ,  r^basis, however, only the sum N = + is diagonal while the differenceis not. This shows that the real ( j f , T|)basis does not provide an h.o. representation with good S3~symmetry properties. A more promising structure, i.e. one m ore in conformity with S 3-sym m etry would be obtained in to m s  o f the following complex combinations,V2 (a,, a* *) =  aj: ±  i arj ; >/2 (a* f , aj *f =  aj= + T i a„ + (3)satisfying the com m utation relations[a j , aj +] =  [aj * , aj *f] =  6 y ; [aj , aj *] = 0  In the new com plex basis, the num ber operatorsNc =  a,** aj ; Nc * =  ; N m =  a4 a j* f  ; Nm * =  N ra + =  8j(4)a ; (5)exhibit better symmetry properties m anifested by the separate S 3~singletsN  = N C +  N c * =  Njj +  Nt, ; N , «  Nc -  Nc * *  (6>Na is thus a new quantum  number having no counterpart in the (^ , tj)  basis. Its eigenvalues, m odulo 3, are a m easure o f departure o f a given state with total quantum  num ber N from a fully sym m etric (antisym m etric) state. [The situation is analogous to the djagonality of the charge operator for a scalar field when expressed in com plex basis (0 , 0  ), but not in the real basis, while the energy remains diagonal in both.] Thus the charge for the complex scalar field seems to play a role analogous to N a in the three-body problem  when viewed in the complex h.o. basis, a  feature which clearly brings out the superiority o f  the complex basis over the real one.S O  ( 2 ,1 )  A lgebras o f  B ilin ea r O p e ra to rsTo consider the SO(2, 1) algebras o f two-step bilinear operators m ade out o f the set (3), it is convenient to distinguish them in accordence with their S3 -sym m etry  properties as followssym m etric : A  =  2 aj aj * ; A + =  2 aj  ^ aj (7)m ixed-sym m etric :C = aj aj ; C* = aj * aj * ; Cf = ajf ajf ; C*f *  aj *+ aj *f (8)W e can identify three distinct sets o f commuting SO (2 , 1) algebras whose spectra are bounded from below. O f these, the S3-sym m etric set (A, A * , N+3) satisfy the commutation relations[A ,N ] -  2 A , [A* ,N ]  =  -  2A+ , [A, A +] -  4  (N+3) (9)so that the three norm alized com ponents areQ+ =  A f / 2  ; Q_ =  -  A /2  ; Q3  =  (N + 3)/2A  n e w  q u a n tu m  n u m b e r  f o r  q q q  b a r y o n s 385and the corresponding C asim ir isu (iH -l) =  -  (AA+ +  A fA ) / 8  +  (N+3) W  ( 1 0 )Since this spectrum  is bounded from below [2], the eigenvalues o f Q 3 are -  u + k, k = 0 , 1 , 2 . . on the one hand, and (N+3)/2 on the other. Thusu (u + l) =  3/4 (even N ) ; + 2 (odd N). ( 1 1 )Similarly, for the m xed sym m etric set C, c \  Nc satisfying[C, N J  = 2C , [Cf , Nc] = -  2 C f , [C, C f ] = 4 (Nc + 3 /2 ) , ( 1 2 )the corresponding C asim ir isuc (uc+ l)  = -  (CC+ + C +C ) /  8 + (Nc+3/2)2/4 (13)whereuc (uc+ l)  = -3 /1 6  (even Nc) ; 5/16 (odd Nc). (14)Identical results hold for starred counterpart (C* , C** , N c*).Finally, the antisym m etric set (N m, Nm* N a) satisfies a normal SO(3) algebra[Nm, Na] = 2 N m , [ N j  , Na] = -  2 N mt , [Nm, Naf ] = -  Na (15)with the C asim irs (s+ l) = (Nm Nmt + Nmf Nm) / 2  + N a 2  /  4 (16)so that spectra is bounded ffrom  both below and above, ju st like (spin) angular momentum.- s £ N a < s ; s  = N /  2 (17)Three-Quark Wave FunctionAs to the signatures o f the new quantum number, N a, (Sec. 1) consider a quark-level three-body problem , viz., the baryon which, after taking out the antisymmetric color part (CoPY /  V6 ), has a sym m etric wave unction in orbital (y ) , spin (x), isospin (0 ) degrees of freedom. In the sam e phase and normalization as eq .( l)  for the complex quantity Zj, we denote ( y c ; y c ) and ( y s ; y j  as the respective doublet and singlet representations of S3  w.r.t. the perm utation symmetry, and the same notation applies for the x. 0- In this notation, the various SU(6 ) states are expressed as|5 6 > q =  y*X*0s ; 156>d -  y *  (Xc0c * +  Xc * 0 c ) / ^  (18)170>q =  X*(Vc0c * +  Vc *0 ,^ /2  ; 170>d =  (yc)^0c +  V c X ^ c ) ^  0 9 )120>q = y“x*0* ; 120>d = y* (Xc0c -  Xc0c )^  (20)Applications to baiyonic spectra are given elsewhere [3].The authors felicitate Prof. Haridas Baneijee on his 60th BirthdayReference*[1] A. N. Mitre, Pkys. Rev. D2S, 1745 (1983).[2] A. N. Mitre, Aqju Slunna a d  B. M. Sodennuk. Pew-Body Sytt. (1995) (in prat)[3] Anju Shanma et. al Pkyx. Rev. DM. 454 (1994).

A New Quantum Number for qqq Baryons

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A New Quantum Number for qqq Baryons

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