Tables of Clebsch-Gordan CoefFicients of SU,P. McNAMEE, S. J.* and FRANK CHILTONtInstitute of Theoretical Physics, DePartment of Physics, Stanford University, Stanford, California

INTRODUCTION

The group SU3 has lately been of considerable interestin regard to the description of elementary particlesymmetries. ' ' Especially a crucial experiment, theGnding of the 0, seems indicative of the relevanceof SU3.' .

Our original motivation in compiling these tables ofthe Clebsch —Gordan coefficients of SU3 resulted fromthe observation that many of our calculations on ele-mentary particle processes which required some labor,if attacked directly using tensorial or projection opera-tor methods, turned out to be almost trivial if one hadalready a complete set of tables of the Clebsch —GordancoeKcients. For this reason, and since there are no otherpublicly circulated tables, we publish our tables here,hopefully in a form which permits efficient use. Ourpurpose in the discussion below is just to present thetables with a summary of the important properties ofthe coefficients necessary to using the tables. For amore detailed discussion of the properties of theClebsch —Gordan coefficients of SU3 see the review ofde Swart. 4

TABLES

where N=dimension of the representation, I'=hyper-charge, I=total isospin, I3——Z component of isospin.The tables are in the form

~ ~ ~

e ~ ~

~ ~ ~

~ ~ ~

QZ$3 P $ ze

and each entry is the (Clebsch —Gordan) coefEcientfor the decomposition of

~ N, 7', I, Is) into~n y i is) ts

TABLE I. Appropriate signs for reversal of order of the states.

1 8 8' 10 10 27 27' 35 35 64

These tables were constructed primarily from theisoscalar factors given by de Swart. 4' The tables wereextensively checked using the fact that each submatrixis unitary. The 888 tables were also checked againsta table constructed by Oakes' using tensorial methods.Spot checks were also made using U-spin and V-spinoperators.

Our states are labeled in conformance with theusage of SU3 as an elementary particle symmetry,

~ N, F', I, Is),~ National Science Foundation Predoctoral Fellow.t This work was supported in part by the U. S. Air Force

through Air Force Once of Scientific Research Contract AF49(638)-1389.' For a discussion of the relevance of SU3 to elementary particlesymmetries see M. Gell-Mann, California Institute of Tech-nology Report, CTSL—20 (unpublished) and Phys. Rev, 125,1067 (1962);Y. Ne'eman, Nucl. Phys. 26, 222 (1961).' For a review of the properties of Lie groups aimed at their usein elementary particle symmetries see R. E. Behrends, J. Dreit-lein, C. Fronsdal, and B.W. Lee, Rev. Mod. Phys. 84, 1 (1962).' V. E. Barnes, P. I. Connolly, D. J. Crennell, B. B. Culwick,W. C. Delaney, et cl., Phys. Rev. Letters 12, 204 (1964).

J. J. de Swart, Rev. Mod. Phys. 35, 91.6 (1963).In preparingour tables, three misprints-in de Swart's tables were found andcorrected.

5 There are other tables of isoscalar factors available, to men-tion: A. R. Edwards, Proc. Roy. Soc. (London) A268, 567(1962);M. A. Rashid, Nuovo Cimento 26, 118 (1962);Y. Dothanand H. Harari, Israel Atomic Energy Commission Report IA—777(unpublished). The reader should be warned of the fact thatconventions for SU3 are not yet standardized and none of thesets of conventions in Refs. 4 and 5 entirely agree.

6 R. J. Oakes (private communication).1

88 1

810

1010 —1

1 —1 —1 —1

1 —1 —1

~

n' y' i' is' ). Tables are presented here for the followingClebsch-Gordan series

Table II. 8 8=270+100+ 100+ 8Q+ 8'0+ 1

Table III: 81310=350+270+10Q+8

Table IV: 827= 640+380+3~0+270+27'0+100+100+8

Table V: 1010=640+270+80+1.Where there are two representations, e.g., 8 and 8',in an outer product we have distinguished them, as isconventional, by R symmetry. '

SYMMETRY PROPERTIESThe complex conjugate of a state can be determined

from( N lr I Is) = (—i)re+"ts

( 8 —ir I —Is).Often, e.g. , in calculating crossing-matrices, one isinterested in the Clebsch —Gordan coefficient with thestates in reversed order. This reversal of order intro-duces a sign of &1. Table I lists the relevant signs.

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1024 REVIEWS OV MODERN PHYSICS ~ OCTOBER 1964

PHYSICAL STATES

The states we use enjoy the standard (Condon andShortleyr) SUs phase conventions. This means that notall of the mathematical states will have the phaseswhich would be most convenient for SU3. For example,it would be convenient for the assignment of the pseudo-scalar mesons to an octet to be such that the octetwas self-adjoint. But because of our convention for thecomplex conjugate state there will be some minus signswhich enter. These minus signs would be compensatedif we made the following assignment of physical statesto our formal states:

j E+)=i81-'-')

)~ )= [80 1 —1), )n')= [80 1 0),

)~+)=—

[ 8 0 11), I ~)= [8 0 0 0),

there are a number of minus signs which occur when1 0 0 0) is written as a sum of products

~8 y i is)8

8 —y i —is) T. hese minus signs would also be com-pensated by the assignment above, making the unitarysinglet a completely symmetric combination of thephysical states. Since under the usual conventions forSU3 tensors the unitary singlet is also completely sym-metric, we see our assignment above is just the rela-tion between our SU2-convention states and the usualSU3 tensors.

For many applications it is more convenient to simplymake the assignment of physical states directly to themathematical states with all plus signs and to rememberthat the octet, indeed even the SUs triplet (rr), is notmanifestly self-adjoint.

Note added im Proof. Dr. P. Tarjanne has kindly in-formed us of his tables of isoscalar factors and Clebsch-Gordan coefficients available as Carnegie Institute ofTechnology Reports NYO —9290 and NYO —9290A (un-published) .

This difference of assignment is more clear if we ex-amine the unitary singlet resulting from the product8138. As the appropriate column of Table II shov s,

7 E. U. Condon and G. H. Shortley, Theory of Atomic Spectra(Cambridge University Press, Cambridge, England, 1935).

ACKNOWLEDGMENTS

We thank Dr. R. J. Oakes for a copy of his tablesand for several discussions. Ke are especially indebtedto Mrs. Dorothy McGrath for her assistance in pre-paring the tables for publication.