On a mass formula without symmetry breaking for SU(n)

[ 8.A--~.5

Nuclear Physics 79 (1966) 645--651; (~) North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher

ON A MASS FORMULA WITHOUT SYMMETRY BREAKING FOR SU(n) J. FORMANEK

Faculty of Technical and Nuclear Physics, Prague Received 18 October 1965 Abstract: A Lie algebra containing the Poincar~ algebra and the intrinsic symmetry algebra SU(n) is constructed. A mass formula for a given type of representations of the algebra is obtained. The parameters in this formula for different irreducible representations are interconnected by relation (19). The sign of the isospin splitting in the baryon decuplet is predicted to be the same as in the baryon octet.

1. In~oducfion A m e t h o d o f constructing infinite-parametrical Lie groups G containing the Poincar6 group and a given g r o u p o f intrinsic symmetry S as subgroups was developed in ref. 1). Masses o f states belonging to the same irreducible representation of G are generally different. The m e t h o d was applied in the cases S = SU(2) (ref. 1)) and S = SU(3) (refs. 2, 3)). The f o r m o f G when S is an arbitrary semi-simple Lie algebra was given in ref. 4). A mass formula for S = SU(2) and S = SU(3) was derived in ref. 4). The case S = SU(n) is studied in this paper. A mass formula for a definite type o f irreducible representations is obtained. It is likely that at least all physically important representations are o f this type. The m e t h o d used for derivation o f the mass formula differs completely f r o m the one used in ref. 4). The formulae obtained for the special cases n = 2, 3 are less restrictive than the similar ones in ref. 4).

2. Construction of the Algebra Let us have the algebra SU(n). (Its order is N = n 2 - 1 and rank l = n - 1 . ) We can construct a fundamental n-dimensional representation o f the algebra in the Hilbert space o f n quarks, the masses o f which (m~, i = 1. . . . n) are in general all different. We choose the first N - l generators as follows X ( 2 , - 1 ) ~-

x(2r)

X (i'k),

~_ ytt, k),

1 =( i ( k =( n, r = 1 . . . . . ½ ( N - l), 645

(1)

,646

~. rom~,mK

•where X (''k> ~ ½(Aikli>
(2)

y(t,k) = 1 (Aikli>
m i-

Ct,k ---- - -

m k

(3)

mk

The relation between r and (i, k) is such that larger r is connected with larger (i, k). We say (j, m) > (i, k) i f j > i or j = i, m > k. The operator A was defined in ref. l) and has the following form:

m=O

m[

where K(a) is a normalization function and the operators xp, ~ , fulfil the commutation relations

[x~, x,] = [ ~ ,

~ , ] = O,

[x~, ~ , ] = --ig~,.

It is easy to see that for any given (i, k) the operators (2) together with Z (''k) = ½([i>
(4)

form a basis of a SU(2) algebra t. Because only n - I elements (4) are linearly independent, we can choose without loss of generality the remaining l generators of SU(n) (forming the Cartan subalgebra) as X (~) - Z (l'k), ~t = N - l + k - 1 . (5) For operators (1) and (5), the following commutation relations hold: [X ('), X (~)] = iC~tJX(~'),

a, fl = 1. . . . N.

(6)

A reducible representation of the Poincar6 algebra in the considered space is realized by means of the operators /I

P~ = ~ , ~ [J>
n

L,,,, = (.~k,,,+Sk,,,) ~ [J>
~k

(7)

~ o + mj s . . ) IJ>
and skin are spin matrices of the quarks. t This is only a special case of the well-known theorem which asserts that any simple Lie algebra is a non-trivial coupling of ½(N--I) algebras SU(2) the basic elements of which are E+~, ~irt(a)//i.

MASS ]FORMULA

647

In order to obtain a set of operators closed under commutation we must add to

(1), (5) and (7) a denumerable infinite number of operators X(:) PI...

~ = 1.

Pm ~

.

.

Ni.

.

m. =. 1,2, .

(8)

In the following we shall formally write the operators (l) and (5) as the operators (8) for m = 0. The commutation relations between all considered operators are then [P~,, P.] = O,

i(guoP.--g,.Pp). [Lu.. L,.o] = i(g.p L , o - g.o.Lup + g , o . L . , - g,p L..).

[P~,, Lov] =

[p~,, v(t.k) "1 = fxpl...pm_ I

(9)

i a O ' k ) v g'k) -~

~llpl...pm}

[e~,, Y(~[' k)p,A = - i~1"" ~)Y", k) "~

• • •

='PP

1 •

• P m

•

~

Iv --I1 ' 7~ ~ P l l,k) • • • PmJ1 = 0 , I-L,, :J Y(~)..I,,,] = i ~ (~ v(~) "~Pt. \YvpjApl*..I

-

~ ...prn--~JltpJ

jr1

X p(~) .....), l-..V.

(10)

Pm

(11)

where a(i,k)

=

rot_ink"

Finally

[Y(=) ~*Px..

•

pn~ *(P) "*Pl...

o,.] = iC~X~(v) " ~

"'Pl

• • •

The relations (9)-(11) define a denumerable infinite dimensional Lie algebra G. The representation of operators (8) in the Hilbert space considered can be obtained as follows. We know 1) that the following commutation relations hold: Ee., A,k] = -

O~kiPpAik.

(12)

Hence

[Pp, X (,,k)] = _ ½Pp(~kiAik[i)(k[ + ~,k Ak,lk)(il).

(13)

If we introduce the four-velocity operator

Up = Pp ~ 1 IJ)
(14)

mj

jr1

we can rewrite eq. (13) as t

ia O.k)V.y(,, ~).

[ p . , X(,. k)] =

(lY)

¢ Let u s define X(i'lO + iy(i,k)

_~ E ± ( i , k )

Relation (13') gives

I'Pp. E±(~'k)-I

----

-t-a°'~)UpE ±(~'k).

(footnote continued o n next page)

648

~. e o e a ~ :

F r o m formulae (13') and (11), one obtains

yp t',k)= Up Y t~,k.)

(15)

Similarly we find that in the considered representation Xf~)

1"41 • • • P m

= U

I~1

• • • [~m

X (~)

where {'1i~I U,,...,m=

U,,

for "for

m = 0 m >0.

q=l

Representations of the form (15) were considered in ref. 4). It is immediately seen from (10) and (12) that the algebra G contains as subalgebras SU(n) and the Poincar6 algebra. The operators (7) and (15) form a representation of G which contains the fundamental representation of SU(n) as the only representation of the internal symmetry sub-algebra. Naturally two questions arise: (i) Do any other representations of G exist? (ii) How to construct the operators of G in such new representations? In the following we prove: There is a class of irreducible representations of G for every irreducible representation of SU(n). We give a prescription of how to construct the operators and the mass formula for these representations. We confine ourselves to representations where the operators which belong to the SU(n) sub-algebra connect ohly states with the same four,velocities. Let us consider a R dimensional irreducible representation D ( 2 1 , . . . 2z) of SU(n). This representation can be realized in the Hilbert space of R particles, the masses of which are the eigenvalues of the mass operator M = a + ~ bjZ tl'~).

(16)

y=2

All irreducible c-number representations of SU(n) are well-known. We can always choose these representations in such a way that the matrix (16) is diagonal. The operators X t~) in the considered Hilbert space one obtains from the corresponding cnumber matrices multiplying elements in the rth row and the sth column by the

Operator A was constructed in such a way that it connects only states with the same four-velocity. Hence

[ u p , x <~)] = 0. This relation can be easily proved also directly. Therefore [ M , E+tt'k)'l = q-a~'~)E +el'k)

must hold. Relations equivalent to the last three were first published in ref. 6).

MASSrORMtrLA

649

operator A(~,~); here ?,~ ~ M , / M ~ - 1 and M , ( r = 1 . . . . R ) is the eigenvalue of M being in the rth row. The generators of the Poincar~ group can be again written down in the form (7) where mj is replaced by Mj, n is replaced by R and skin are spin matrices of particles considered now. Similarly one obtains the operator of four-velocity in this representation. Let us define X ~(~) . . . ~ , , __- h " U ~ . . . ~ , , X

m = 0, 1. . . . .

(~),

(17)

where h is an arbitrary real number and X (~) the operators whose construction has just been described. It can be easily shown that we have obtained an irreducible representation of G if and only if bj = h

m j-

n k

rn k

,

j = 2 .....

n.

(18)

Therefore we are able to construct a class of irreducible representations of G for every irreducible representation of SU(n). Different representations of G belonging to the same class are distinguished by the values of the spin and the parameter h (i.e. they contain different representations of the Poincar6 group). It is seen from formula (18) that for every representation of G, the following relations hold: bi _ bk bj- bm -

a(~, k) a (i'm) "

(19)

In particular, for S = SU(2), one obtains Z(1, 2) = T3, the operator (16) is M = a + bT3,

(20)

and no relation among coefficients a, b in different representations is obtained. For S = SU(3) we have Z(1, 2) = T3,

Z (1'3) _- ¼ Y + ½ T 3 ,

and the mass formula following from (16) is t M

= a+bY+cT3.

(21)

Formula (16) asserts that coefficients b and c in different representations are connected through the relation b

at3, 1)

C

a(2, l) -½"

(22)

t W e see that M commutes with T ~. Hence the mass formula (21) allows classification of particles according to Ta, Y and T ~.

650

j. FORM~NEK

The mass formulae ( 2 1 ) a n d (22) were obtained already in ref. 3) for the special case of representation D(1, 1) of the sub-algebra SU(3). It follows from this method 3) that other representations of G containing D(1, 1) of SU(3) do not exist (if SU(3) sub-algebra connects only states with the same U~). It is likely that the same is true also for S = SU(n) for any n. The mass formulae (20) and (21) were obtained also in ref. 4) by a completely different method. The constants b and c in (20) and (21) are 4) the same for all multiplets. This is a simple consequence of the fact that only a subset of the representations considered here was used in ref. 4). It was shown 3) that the parameters a, b and c could be chosen in such a way that the formula (21) gives (besides several better relations) the mass spectrum of the eight baryons belonging to the octet with deviation from experiment smaller than 3.5 ~o. The well-known equidistant rule for the mean masses of isomultiplets in the decuplet follows also from (21). Moreover the relation t N * - - N* + + = ½(Y*- - Y* +) following from the formula (21) is not in disagreement with experimental values 6) N * - - N *++ -- -0.6___5 MeV;

Y*--Y*+

= 4.4___2.2 MeV.

Using the relation (22) one obtains that c for the decuplet (as well as for the octet) is negative. This prediction agrees within the experimental errors at least for the aforementioned resonances. According to the relation (22) parameter c would be approximately three times larger for the decuplet than for the octet. The considered experiments give a smaller value for c. If we remember that the algebra used above does not explain the mass difference between A and Eo, we cannot be too surprised that neither in this point the quantitative agreement with experiment is very good. 3. Conclusion An algebra G containing S = SU(n) and the Poinear6 algebra as sub-algebras has been given in sect. 2. A definite class of irreducible representations of G has been obtained and the corresponding mass formula written down. It is clear that we can proceed similarly also in the case when S is an arbitrary simple Lie algebra. Especially we can assert that also in this general case the mass formulae are linear in l members of the Cartan sub-algebra. It is worthwhile to note that the parameters of the mass formulae (for a given algebra G) are connected by means of relations similar to (19). Hence one obtains non-trivial relations among masses of particles belonging to different multiplets. The consequences of this for baryons have been discussed at the end of sect. 2. t This relation follows also from the Matthews-Feldman parallelogram rule which is automatically fulfilled by the mass formula (21).

MASS FORMULA

651

If all representations of G are of the considered type then the algebras so far studied cannot explain non-degenerate mass spectrum of mesons and, for baryons, they cannot explain different masses of particles belonging to the same supermultiplet and having the same T 3 and Y but different T 2. The formal reason for these degeneracies is that the first three relations (10) apparently cannot be fulfilled if the mass operator contains terms bilinear or terms of even higher order in operators belonging to SU(n) sub-algebra (excluding Casimir operators of the SU(n)). The natural question arises how to generalize the considered algebras in order to allow higher terms in the mass formulae. We shall not discuss this here and devote a special paper to it. We would like to stress once more that the algebras G considered so far do not contain operators connecting states with different spin. Therefore all particles belonging to one irreducible representation of G have the same spin. On the other hand if there is an irreducible representation of G containing particles with a given spin, then a similar representation exists in the space of particles with any other spin. Spin independence of the mass formula is a special consequence of it. The principal possibility of connecting spin and intrinsic degrees of freedom in the framework of the considered algebras has been stressed 2). One type of such connection is given in the following article. The author wishes to thank Professor Dr. V. Votruba, M. Havli~ek and J. Votruba for constant interest in this work. The author is also indebted to Professor Dr. V. Votruba for calling his attention to ref. 6). References

1) 2) 3) 4) 5) 6)

J. Form~inek,Czech. J. Phys., to be published J. FormAnek, Czech. J. Phys., to be published J. Form6nek, Nuovo Cim., submitted J. Votruba and M. Havli~ek, Nuovo Cim., submitted J. Werle, Nuovo Cim., submitted 1964 Int. Conf. on High Energy Physics, Dubna, August 5-15, Dubna, preprint P-1887, p. 19