Constraints on the statistical tensor for low-spin particles produced in strong interaction processes

[

~

NuclearPhysics 87

(1966) 8 9 - - 9 9 ; (~)

North-Holland Publishing Co., Amsterdam

Not to be reproduced by photoprint or microfilm without written permission from the publisher

CONSTRAINTS ON T H E STATISTICAL T E N S O R FOR L O W - S P I N PARTICLES P R O D U C E D IN S T R O N G INTERACTION P R O C E S S E S R. H. D A L I T Z

The ClarendonLaboratory, Oxford Received 16 M a y 1966 A b s t r a c t : T h e r e q u i r e m e n t that all the eigenvalues o f the spin density matrix p are positive imposes M which characterize p. a set of constraints on the values possible for the statistical tensors tL

Here, these constraints on t~ are given explicitly for spin values l ~ 2 for the case of particles produced in strong interaction processes. The constraints on p itself are given as a set of inequalities, valid for arbitrary/. 1. Introduction

The use of the spin density matrix p has become a very common technique for the analysis and interpretation of data obtained on the production and decay of resonant states produced in strong interaction processes (for convenient reviews, see Jackson 1) and Dalitz 2)). For a resonant state c* of spin J, the density matrix p is conveniently specified by the statistical tensors t~, the explicit expression for p then being as follows (cf. Byers and Fenster 3)): 2J

L

P = ( 2 J + l ) -1 E ( 2 L + 1 ) ~ L=O

t~*Tff(J),

(1.1)

M= --L

where TM(j) denotes the spin tensor constructed from the spin operator J to have the same transformation properties with respect to rotations as do the spherical harmonics yM. The operators TM(j) have the reality property that

TM(j) t = ( - 1)MTL~t(J).

(1.2)

They are normalized such that

(Jmlr~(J)[Jm') = (Jm 'LM[JLJm).

(1.3)

Since the density matrix is hermitian, the statistical tensors t~ have the corresponding reality property that t~* = ( - 1)Mt;. M. (1.4) We consider generally a c* production process of the type a + b --+ c * + d + e + . . . .

(1.5)

where observations are made only on the state c*, an average being taken over the 89 Oktober 1966

90

R.H.

DALITZ

production directions and decay configurations of the accompanying final particles, d, e . . . . The c* density matrix can then depend only on the m o m e n t u m vectors k and k', the momenta of a and c* respectively (specified in the c.m. frame for the initial system), and can necessarily be written in the form 2J

p = ~_, ( J . n)SFs(J • k, J . k'),

(1.6)

s=0

where n denotes the unit vector parallel to k × k', normal to the plane of k and k'. As pointed out by Capps 4), parity conservation for the production interaction requires that the functions F~ be invariant under space reflection, i.e., for k ~ - k and k' ~ - k'; hence the F~ are even functions of J • k and J • k'. When the quantization axis is chosen along n, this property leads to a "checkerboard pattern" for the matrix Pm~, = (JmlplJm'); since the operators (g • k) 2, J • k J • k', and (J • k') 2 allow only Am = +2, 0, and J • n allows only Am = 0, the matrix elements p~.,, for expression (1.6) are then zero except for the elements such that ( m - m ' ) = even. This conclusion (due to Capps 4)) naturally requires that all the statistical tensors t~ with M = odd are necessarily zero. With quantization axis n, then, the matrix P~m, consists of two interpenetrating matrices. For J = integer, these two submatrices are p~, the submatrix of odd dimension ( J or J + 1, whichever is odd) formed of the P,,m" with m and m' even, and Po, the submatrix of even dimension formed of the p,,,,, with m and m' odd. For J = halfinteger, the two sub-matrices are p+, the submatrix of dimension ( j + l ) consisting of the elements p ~ , for which m and m' differ from J by an even integer, and p_, the submatrix of dimension ( J + ½ ) consisting of the elements p~,,, for which m and m' differ from - J by an even integer. Under the operation of time reversal, we have

J--+-J,

k ~-k,

k'~-k'.

(1.7)

This operation leaves the functions Fs unchanged and reverses J . n. Its effect on p thus corresponds to the effect of simply time-reversing the spin operator, namely J ~ - J . Time reversal of the spin has the following effect on p: (i) t ~ r k - - l~Lt M ) L, (ii) J • n -~ - J • n, and the density matrix elements undergo the transformation Prom" ~ P-re',-,," In other words, in terms of the t~, the element p_m,, -,, may be obtained from P,,m" by (i), that is by reversing the sign of the t~ with odd L. For J integral, time-reversal of the spin leaves Pe and Po invariant in form; the eigenvalues of Po and pe are then generally independent. For J half-integral, time reversal of the spin interchanges p+ and p_, thus p+ ~ p_ ; the eigenvalues of p_ are then obtained from those of p+ by reversing the signs of all t~ with odd L. For an assumed c* spin value J, the statistical tensors appropriate to the state c*

CONSTRAINTS ON STATISTICAL TENSOR

91

may be deduced, at least in part, from the c* decay angular distributions. For mesonic resonances with decay m* ~ P + P (where P denotes a pseudoscalar meson), only the tensors t~ with L = even can be deduced from the observed distributions; for decay processes m* ~ P + V (where V denotes a vector meson) the deduction of all t~ for even L generally requires a knowledge not only of this decay angular distribution but also of that for the subsequent V decay and of its correlation with the parent decay process. For baryonic resonances B* undergoing the decay B* ~ B + P, the situation has been discussed in detail by Byers and Fenster 3). When a complete determination of the polarization of the final baryon is possible (for example, when B denotes a A hyperon, since the decay process A ~ p + ~r- provides an efficient polarization analyser for the A hyperon), the t~ with L = odd may be deduced from the polarization angular distribution in two independent ways. For each non-zero t~ with odd L, the comparison of the two estimates obtained provides an independent determination for both the spin J and the parity w for the state B*. The t~ for L = even may be deduced from the B* decay angular distribution. When there is no B* polarization (all t~ = 0 for L = odd) or a polarization analysis for B is not possible, the t~ for L = even still allow limits to be placed on the B* spin value J. Assuming the contribution of non-resonant, background to the observed decay angular distribution to be negligible, the spin J is necessarily larger than the values ½2 corresponding to all the non-zero statistical tensors t~. On the other hand, the spin J must always satisfy the Eberhard-Good inequality 4) 2J

2J+l

=< Q Z ( 2 L + l ) L=O

L

Z

tL M

2,

(1.8)

M= -L

where Q denotes the number of incoherent pure spin states needed to characterize all the particles other than c* in the production reaction (1.5) (for example, for the baryonic resonance B*, Q = 2 for processes of the form PB -~ PB* since the target baryon (necessarily a proton) nas only two independent spin states, whereas Q = 6 for processes of the form PB --* VB*). Further constraints on the statistical tensors t~ exist in consequence of the probabilistic character of the density matrix. If the eigenvalues and eigenstates of p are denoted by p~ and qS~, respectively, for ~ = l, 2 . . . . ( 2 J + 1), then a special mixture of states leading to p is provided by the set of spin states ~b~, each occurring with probability p~. This necessarily requires that these eigenvalues p~ are all positive. This requirement of positive definiteness for p leads to a complicated constraint on the statistical tensors tL u which has not been generally recognized in the literature * It is possible that this constraint could lead to the rejection of some spin values J for which the t~ obtained satisfy the constraints already mentioned above. In the note, it is our purpose to obtain explicit expressions for these constraints for the low t S i n c e t h e c o m p l e t i o n o f m o s t o f t h e w o r k d i s c u s s e d in this p a p e r , t h e r e h a s a p p e a r e d a d i s c u s s i o n o f t h i s c o n s t r a i n t for t h e case J = 1 b y M i n n a e r t 5). W e h a v e t a k e n a d v a n t a g e o f t h e e l e g a n t p r o c e d u r e underlying his discussion in developing part of our discussion in sect. 3 below.

92

R.H. DALITZ

spin values J < 2 and to indicate explicitly the nature of these constraints for higher spin values in terms of the density matrix. 2. Constraints Resulting from Positive-Definiteness for p 2.1. THE J = ½ CASE

For this case, the situation is well known. With quantization axis n, the density matrix p is necessarily diagonal. With expression (1.1), the eigenvalues are explicitly ½(1 +~/3t°). These are both positive as long as ito[ < __1

(2.1)

Of course, for this case, t o is simply PIll3, where P denotes the polarization of the state, and eq. (2.1) states simply that the polarization cannot exceed 100 ~o. 2.2. THE 3"= I CASE

With quantization axis n, the density matrix has only the following non-zero elements: P

= P-l,1

Po0

()

0

P-~,-1/

,

with p_ ~, 1 = P*, - 1 and the condition of unit total probability Tr(p) = P l l + P o o + P - 1 , - x = 1.

(2.2)

With expressions (1.1) and (1.3), these elements are related to the statistical tensors by the expressions P ll = ½+ x/{rt° + ~x/~t2° , (2.3a) P--l, --1 = ~ - - ~ t l 0 "]-~ 4 ~ t 0 '

(2.38)

m , - 1 = 4 -~z t2.

(2.3c)

We note that p_ 1,-1 may be obtained from Px 1 by reversing the sign of t °, as expected from the general discussion above. The equation for the eigenvalues p= necessarily consists of the product of two distinct eigenvalue equations

{p--Poo}{Pz--P(Pli+P-1,-1)+PIiP-1,-1-IPl,-il

2} = 0,

(2.4)

the first factor corresponding to the sub-matrix Pc, the second factor corresponding to the sub-matrix Po, as expected,. The three eigenvalues of p are therefore {P,} = {Poo, ½(Pll + P - l , - 1 +-~/(Pll-P-1,-1)2+4[Pl,-112)} •

(2.5)

With eq. (2.2), the necessary and sufficient conditions for these three eigenvalues all

CONSTRAINTS ON STATISTICAL TENSOR

93

to be positive is

x/{(P11-P-l,-i)2+41Pl,-112}

<= Plx + P - 1 , - 1 < 1.

(2.6)

With the expressions (2.3), these inequalities m a y be stated succintly t in terms of the statistical tensors, -~/2+~/(9(t°)2+61t212) < t o < ~/~-~.

(2.7)

As discussed in sect. 1, the polarization tensor t o cannot be deduced f r o m the decay data for a mesonic resonance state. In this case, the lower limit (2.7) for t o m a y be regarded as providing an upper limit for t o . Alternatively, the lower limit (2.7) m a y be replaced by the following lower limiti -~/~-+~/glt~l < t o < x/~o.

(2.8)

As long as the observed values for t22 and t ° satisfy (2.8), there is no necessary conflict with the inequalities (2.7). Since Q > 4 for V production in PB collisions, the E b e r h a r d - G o o d inequality (1.8) provides no constraint on the t~; the L = 0 term Q on the right h a n d side then already exceeds the limit ( 2 J + 1). 2.3. T H E

J = ~ CASE

Here the density matrix takes the f o r m

(?

0 P~

p_~,g 0

*

0 p_~,~

)

P-~,~

P-~, -~ 0

p_g, _ g /

.

This matrix separates into two 2 x 2 matrices p+ and p_. The elements o f p + are given explicitly in terms o f the statistical tensors as follows: pg,~ = ¼{1 +3x/-ft,+x/5t2+x/-~t3}, ;o -o 7o x P - ½, - ~ = 4 {1 -

30

43-tl 1

--2

-

-o 7o 4 5 t 2 q- 3x/vt3},

P-g, ~ = ~{x/10tz + x / ~ t 2 } ,

(2.9a) (2.9b) (2.9c)

The elements o f p_ ( p _ g _ g , p~,g and p_g, g) are obtained f r o m the expressions (2.9a, b, c) for pg,g, p_½,_,, and p_~,g, in turn, by reversing the signs o f t °, t o and t 2 in these expressions. The eigenvalue equation of p ÷ is

(P~, ~--P)(P-~, - ~ - - P ) - - I P - ~ , ~12 -- 0, (2.10) t These inequalities are equivalent to the inequalities obtained recently by Minnaert s) for the case of spin one by a more elegant and general procedure. They were also obtained earlier ~) by the pedestrian method followed here, although their final statement in ref. ~) is not strictly correct, as a result of squaring the inequality obtained. Note added in proof." We have learned from Dr. P. Minnaert that results for J = 1 equivalent to these results were published some years ago by Lakin 9).

94

R.H. DALITZ

with the solutions

u± = ½{p~, ~+ p_~, -½+x/((P~, ~ - P - ~ , _,)2 +41p-~, ~12)} =¼{1 + ~/yt 1S° + 2~/_et 3~o -+ x/[(2~/Tt 1~° + ~/5t 2-0 _ ~/_eta)~o2+ (x/10t2--2+ x/~t2)2]}. (2. l la) The eigenvalues v± of p_ may then be written down at once

v± = ¼{1- x / ~ t ° - 2~/~t° + x / [ ( - 2x/3t° + ~/5t2° +~/Yt°'~3, 23+ (x/iOt22- x/i-4t 2)2] }. (2.11b) The necessary and sufficient condition that all the eigenvalues of p are positive are then

u_ > 0,

v_ > 0.

(2.12)

With expressions (2.11), these conditions may be stated succintly by the inequalities: 3 0 + 2X/3-t3 7o -- 1 + x/{(2x/a~t° +~/5t°--x/Svt°) 2 +[x/~t 2 -t- x/i4t2l 2} __
__ 1_4{(_2

/3to +x/5t2+x/yt3) -o -2 - 2 2 }. +l~/10t2-x/14t31

(2.13)

For example, for J -- 3, the investigation of Sharer and Huwe 6) on Y*(1385) decay led to the following estimates for the Y* statistical tensors appropriate to the conditions of their experiment, t o -- -0.05+0.07,

t o = -0.12___0.03,

t o = 0.26-+0.04,

t22 = - 0.02-+ 0.02 + i ( - 0.03 +__0.02), t~ = - 0.105_ 0.035 + i ( - 0.01 _ 0.03).

(2.14)

For these values, the inequalities (2.12) read -0.21-t-0.15 < 0.58-t-0.21 < 0.64-T-0.13,

(2.15)

where the major part of the error specified is that due to the error in t32. The upper inequality (2.15) is therefore satisfied by the best values for the tuL, but is not satisfied for all the values t u possible within the errors. The closeness of the upper inequality for the best values t u implies that one of the eigenvalues of the Y* density matrix in their experiment happens to be rather small; the requirement that this eigenvalue be positive then provides a significant constraint on the t~ values permissible. A particular situation of interest is that in which all polarization tensors (i.e., all t~ for L --- odd) are zero. This corresponds to the situation in which the expression (1.6) for p includes only terms with s = even. In this situation, no information can be obtained about the parity of the state B* from the study of its decay characteristics. For this situation, the inequalities (2.13) reduce to -- 1 + x/{5(t°) 2 -t- 101t2212} < 0 < 1 -x/{5(t°) 2 d- 101t212},

CONSTRAINTS ON STATISTICAL TENSOR

95

i.e. to the single inequality 5(t°)z+ 10[t2l 2 < 1.

(2.16)

For a production reaction of the form P + B -~ P+B*,

(2.17)

we have Q = 2, and the Eberhard-Good inequality (1.8) reduces to 4 < 2{l+5(t°)z+101t212), putting t o = t o = t 2 = 0, i.e., to the inequality

5(t°)2+ 10It212 _-> 1.

(2.18)

Hence, for the common production reaction (2.17), the complete absence of B* polarization would require the equality

5(t°)2+ I01t2212 =

1

(2.19)

to hold between the two statistical tensors not required to be zero in this situation. 3. Discussion

The pedestrian method adopted above for discussing the constraints following from the positive definiteness of p, involving the explicit calculation of the eigenvalues of p, cannot readily be carried further to higher values for the spin J, since explicit solution of the eigenvalue equations takes a complicated form for 3 x 3 matrices and beyond. In order to indicate the procedure appropriate for higher spin values, let us first consider here the case J = 2 in detail. For J = 2, the submatrix Po has dimension 2 x 2, and the structure

)

p-l,~ p-~,-1/= \ ~* ~-P ' where ,

(3.1a)

fl = - 51v/~t° + sl-x/~t °,

(3.1b)

~t = 1 - x / ~ l x t ° - ~ x / ~ t °

~1 =

x/~t "7" 2 2 - - ~ 6 t2"

(3.1C)

The eigenvalues of this matrix are given by the same expression as the last two roots (2.5) of the eigenvalue equation (2.4). The requirement that these two roots be positive is given by the left inequality (2.6). However, the expressions for Pl 1, Pl,- z and p_ 1, in terms of the statistical tensors are different here, and lead to the left inequality 2

-~-~/~rtl + 51-~/14t3) +

v 2-

<

2

,/~_,0

12

/2,0

(

1.

(3.2)

96

R . H . DALITZ

The sub-matrix pe has dimension (3 x 3), with the elements P22 = ~-+ t °, 1 51-~/6t1+ 1 --0 4 ~2_ t o 2 "4- y14 ~ t --03 3t" 3.~-'1",5 PO0

~

1 . / 2 t o2 ~±5 v~~ / 2 t - o4 ,

(3.3b)

~--v~-

2 + p2o = ,/ 2t2

(3.3a)

-i~ 2 + 3,/ - - t 2,

(3.3c) (3.3d)

P2,-2 = 3x/~ t~,

the elements P-2,-2 and Po,-2 being obtained from (3.3a) and (3.3c) respectively, by reversing the signs of t °, t o and t 2 in those expressions. The eigenvalue equation may be written det(1-2p)

=

1-c1~.+c2j.2-c3~. 3

=

0,

(3.4)

where the eigenvalues p, = 1/2, are given by the roots of eq. (3.4). The roo,s 2, of this equation are necessarily real, since the hermitian property of Pe ensures that its eigenvalues are all real. From the Descartes rule of signs (cf. ref. 7), for example), the necessary and sufficient conditions for eq. (3.4) to have no negative roots are that all the coefficients el should be positive c~ _>_0,

i = 1, 2, 3.

(3.5)

Since the roots are all real, these inequalities are also the necessary and sufficient conditions (as is stated by Minnaert 5)) for all the roots of the eigenvalue equation (3.4) to be positive. The coefficient el is simply (P22+Poo+P-2,-2), which is equal to (1-200. The condition cl > 0 therefore leads to the right inequality (3.2), i.e., 2~ < 1. The coefficient c 2 is given by the sum of the second-rank minors down the diagonal of Pc" Explicitly, this leads to the inequality 1 20 (y-3x/vt2 +

334~5 14

t°)(~ +

20 4~-t2+

~ / - - ~ ,0"~ 5vTW~4J

1 --0 1 702 + ({x/6tl + 5!~/zt3)

22 -'3-22 9 42 --2(~/Tt2+3~/~6t4) - - r2l8t a l2 2 +-~lt41 >_- 0.

(3.6)

The coefficient c3 is given by det(Pe). Explicitly, this condition is 1 x/vt2_F 20 9 2 0 1 x/vt2 2 0 + 3x/1_ - -~ to) 2 _(yx/6tl (~_ ~_~/_¢t4){(y.F ) -- a0 q_y~/{ta 1 -- 0 2 _ _ y l t 4 149- 2 )

1 20 3s__x/~_ ~ t ° )(1 x/2tz -3-1t22 3[) --2 (y+4~-t2d" ~- 2+3473~t2[2+14 4 --0 70 1"42" 22 2-~2 +v(x/6q +~/~t3) Re (~/~-t3 (~/vt2 +x/v~t4)) 1 4,* 22 ~-22 + 6~/-s-Re(t4 ((x/vt2 +x/~-,t4) - T14( t z )2 2 ) >= O.

(3.7)

CONSTRAINTS ON STATISTICAL TENSOR

97

In the general case, for spin J, let us consider the complete density matrix p. The eigenvalue equation corresponding to (3.4) is here 2J+l

det ( 1 - 2 p ) = E (-1)"cn2" = 0,

(3.8)

n=O

with c o = 1. The hermitian character of p ensures that all roots 2~ of this equation are real. The Descartes rule of signs leads to the necessary and sufficient condition for all the roots of eq. (3.8) to be positive, namely cn > 0,

n = 1, 2 . . . (2J+ 1).

(3.9)

These conditions have been stated for the case J = 1 by Minnaert s), who has shown that, for this case, the coefficients c2 and c 3 can be neatly expressed in terms of the quantities T, for n = 1, 2 and 3, where T, denotes the trace, T~ = Tr(p").

(3.10)

The general expression for the coefficients c, in terms of the traces Tn may readily be obtained, as follows. In terms of the eigenvalues (p~}, we have 2J+l

2J+l

det ( 1 - 2 p ) = I-[ (1-2p~) = exp { X In (1-2p~)}. g=l

(3.11)

~=1

Expanding the logarithm in powers of 2, the exponent of expression (3.11) becomes -

s=l

-

p~)

=

-

S

s=l

- - , S

(3.12)

since T~ = Tr(p ~) = ~ 2 J +1l ,t p~). 2, From the equality 2.,~= 2J+ 1

E ( - 1 ) ' c . 2 " = I~I {exp (-2"T~/s)},

(3.13)

s=l

n=O

we can deduce the desired relationship explicitly c~

:

( - 1 ) " ~ {s=ITI1( - 1)"*(T-~)~/

sn-(n,)! J'

(3.14)

where the sum is taken over all sets of positive integers {n~} such that ~o= Isns = n. For n = 2, the inequality obtained is well known c 2 = ½ ( r ? - r 2 ) __>0.

(3.15)

For n = 3, the inequality obtained is that given by Minnaert s), c3 = - ~ T 3 - ½ T z T~ +X6T3 > O.

(3.16)

For n -- 4, the inequality obtained is explicitly c4

=

--¼T4+~T 3 T1 + X s T ~ - ¼ T 2 T~2+-~-¢T( > O.

(3.17)

98

R.H. DALITZ

And so on, up to n = 2 J + l . In terms of the tL ~, the inequality (3.15) has the well-known form 3) (since the trace 7"1= 1), 2./

7"2

1

L

~ (2L+l) ~ ItS[ z < 1.

2 J + l L=O

(3.18)

M=-L

For the inequality (3.16), we need an expression for T 3, T3 = ( 2 J + 1) -3 ~'. (2L+ 1)(2/+ 1)(2;t+ 1)t~*t'r*tuz * Tr (TLMTzmT~).

(3.19)

Ll~.Mmlz

Using the expression (1.3) for the matrix element of TLM, the trace in this expression may readily be transformed to the concise form Tr(TL~TtmT;)=(2J+I)~

J

J

M

m

p

'

where the curly bracket denotes the 6-j symbol and the round bracket denotes the 3-j symbol, as defined and tabulated by Rotenberg et al. s). Hence, in terms of the tL M, the inequality (3.16) takes the general form 2 ~ (2L+ 1)(2/+ 1)(22+ 1) (2J+l)~z,t,z=o 2J

J

M=-Zm=-I

-4 M

rn #

L

,U**m*,u* > - - 1 + - - 3 E ( 2 L + l ) E ItLM[2" ×'L ~l "Z = 2J+IL=o M=-L

(3.21)

Explicit expressions for the inequalities (3.9) for higher values of n are of increasing complication, and we shall not consider these expressions in detail. We should note again that, with parity conservation in the c* production process, the density matrix p is reducible, into two sub-matrices of rank (J+½) if J is half'integral, into two sub-matrices of ranks J and J + 1 if J is integral. If the general considerations discussed just above are applied to the sub-matrices rather than to the complete matrix p, inequalities polynomial in the t~ are obtained which involve lower powers of the tL M than do the inequalities (3.9) obtained with the use of p itself. For example, with J = 2, the general argument leads to a series of polynomial inequalities of order n = 2, 3, 4 and 5. However, consideration of the submatrices Pe and Po lead to a quadratic inequality and to linear, quadratic and cubic inequalities respectively. This means that, with parity conservation, the set of 2J inequalities (3.9) obtained corresponding to the complete matrix p can necessarily be simplified in form. This feature appears far from obvious in the form of the general expressions (3.10) and (3.14). In this paper, we have carried out this simplification explicitly for all the spin values up to and including J = 2, and it is clear that this pedestrian method of deriving the simplified inequalities can be carried out for higher spin values, by making use of the relations (3.14), the traces Tn, and the inequalities (3.9) appropriate to each of the two sub-matrices of p separately, as far as desired.

CONSTRAINTS ON STATISTICAL TENSOR

99

References 1) J. D. Jackson, Revs. Mod. Phys. 37 (1965) 484 2) R. H. Dalitz, Proc. Intl. School of Physics "Enrico Fermi", Course 33, Strong Interactions (Academic Press, New York, 1966) p. 141 3) N. Byers and S. Fenster, Phys. Rev. Lett. 11 (1963) 52 4) R. H. Capps, Phys. Rev. 122 (1961) 929 5) P. Minnaert, Phys. Rev. Lett. 16 (1966) 672 6) J. B. Shafer and D. O. Huwe, Phys. Rev. 134 (1964) B1372 7) C. V. Durell and A. Robson, Advanced algebra (Bell and Sons, London, 1937) p. 287 8) M. Rotenberg, R. Bivins, 1'4. Metropolis and K. J. Wooten, The 3-j and 6-j symbols (Crosby Lockwood and Sons, London, 1959) 9) W. Lakin, Phys. Rev. 98 (1955) 169