On the Masses of Nonstrange Pseudoscalar Mesons and the Generalized Klein-Gordon Equation. ~. FLAT(), D. STEI~NHEIMER, J. STERNHEIMER and J. P. ...
IL NUOVO CISIENTO
VOL. X L I I A, N. 2
21 Marzo 1966
On the Masses of Nonstrange Pseudoscalar Mesons and the Generalized Klein-Gordon Equation. ~ . FLAT(), D. STEI~NHEIMER, J. STERNHEIMER and J. P. VIGIER $nstitut Henri Poincar~ - Paris G . W A T AGHI~"
i s t i t u t o di F i s i c a dell' U n i v e r s i t d - T o r i n o
(ricevuto il 29 Gennaio 1966)
Some authors treated recently the problem of strongly interacting particles by the use of Riemannian manifolds. One of the aspects of this problem was to consider the cigenvaluc problem for the Laplace-Beltrami operator as a generalized KleinGordon equation. i n the following, we shall show how such a consideration enables us to remove the degeneracy between nonstrange pseudoscalar mesons with essentially the same q u a n t u m numbers, the levels being due to curvature of space. In a very interesting attempt, R.~CZK~ (~) wrote the Laplace-Beltrami operator for the manifold S2x S 1 • R (S n being the ~-dimensional sphere and R the real line), with a given metric. In this case, the corresponding group of m o v e m e n t was Oa • U~ x R . By the aid of these considerations, a , mass formula ~) was deduced w h i c h - - f o r the proper choice of the r a d i i - - c o i n c i d e d with the 0 k u b o formula for pseudoscalar mesons. However, this t r e a tm e n t calls for some remarks: a) Existence of half-integer values of the isospin I is not so clear from the model. b) If we admit the author's point of view, it proves once more (at least in the 0case) that the Okubo formula stands for itself, and is neither dependent on the S U 3 theory nor on the symmetry-breaking process. By interpreting the mass operator as the invariant g ~ P t , P~ of the Poincar& Lie algebra P, two of us (2) ~ e r e led to the problem of connecting nontrivially the external (t)oincard) and internal Lie algebra, and getting therefore mass relations. (1) R. RACZKA: ] ) r e p r i n t Trieste ( M a r c h 1965) IC/65/32. (~) 51. FLAT() ~lnd 1). STERNHEIS[ER: Com~)l. l~e~'*d.. 260. :r
(1965).
43"]
M. FLA.TO, D. S T E R N ' t I E I ~ E R ,
J.
STERNHEIMER,
g . P.
VIGIFR
and
G. ~ r A T A G H I N
T w o special cases w e r e t h e n c o n s i d e r e d : T h e c o n f o r m a l L i e a l g e b r a S U2.~, as a u n i f i c a t i o n of P (or S04,1) a n d SU~. 1 t a k e n as t h e i n t e r n a l L i e a l g e b r a (3), a n d t h e u n i f i c a t i o n of P a n d S L ( 3 C ) . T h e r e s u l t s w e r e (4) i n p e r f e c t a g r e e m e n t w i t h e x p e r i m e n t a l d a t a for b a r y o n s . H o w e v e r for mesons, t h e r e r e m a i n e d t h e p r o b l e m t h a t t h e i r m a s s e s a r e n o t e n t i r e l y c h a r a c t e r i z e d b y t h e q u a n t u m n u m b e r s (JP, I , _ra, Y). I t w a s t h e n s u g g e s t e d p h e n o m e n o l o g i c a l l y (4) to i n t r o d u c e a k i n d of p r i n c i p a l q u a n t u m n u m b e r in o r d e r t o r e m o v e t h e d e g e n e r a c y . W e s h a l l n o w see t h a t s u c h a q u a n t u m n u m b e r c a n a r i s e n a t u r a l l y f~om t h e c o n s i d e r a t i o n of a c u r v e d space, t h e g r o u p of m o v e m e n t of w h i c h c a n b e c o n s i d e r e d as a , d e g e n e r a c y - r e m o v i n g g r o u p ~). I t is well k n o w n t h a t t h e m a x i m a l o r d e r of t h e g r o u p s of m o v e m e n t of a R i e m a n n i a n n . m a n i f o l d V~ is 89 + 1) (5), t h e m a x i m a l o r d e r b e i n g o b t a i n e d for c o n s t a n t c u r v a t u r e s p a c e s (spheres, p s e u d o s p h e r e s or flat spaces). I n o r d e r to h a v e a c o h e r e n t m a s s s p e c t r u m , w e s h a l l c o n s i d e r a 174 as S * • ( t h e so-called E i n s t e i n model), w h i c h c a n b e e m b e d d e d i s o m e t r i c a l l y in R 6 as a h y p e r s p h e r i c a l c y l i n d e r , S 3 b e i n g t h e h y p e r s p h e r e z~ + z] 4- z~ 4- z~ = r 2 i n /;#(Zl . . . . . z,). W e t a k e o n V4 t h e m e t r i c : ds~ = dt 2 - d a z w h e r e d a z is t h e n a t u r a l m e t r i c of 8 a i n t h e f o u r - d i m e n s i o n a l E u c l i d e a n s p a c e Ed. N a t u r a l l y t h e g r o u p of m o v e m e n t of o u r V4 is 04 • R. One d e d u c e s t h e r e f r o m t h e L a p l a c e - B e l t r a m i o p e r a t o r , D=~z/~tz--(1/r'2)l-~, w h e r e /~ is defined b y r'~ A = (1/r')(~/~r')(r's(~/~r'))+ I~ (A b e i n g t h e u s u a l L a p l a c l a n i n E~). W e d e n o t e b y P a n - t h - o r d e r h o m o g e n e o u s h a r m o n i c ( A P = 0) p o l y n o m i a l , a n d define t h e h y p e r s p h e r i c a l f u n c t i o n s of d e g r e e n, Y . , b y P(z~ ..... zd) -- r" Y . . Y~ t h e n satisfies t h e p a r t i a l d i f f e r e n t i a l e q u a t i o n / i Y . + n ( n + 2) Y. =- 0. I n a d d i t i o n o n e c a n show t h a t
f
l r . :F,, e d/2
0
for
q~ -~ n ~ .
88
F o r a fixed n, we h a v e
= (n + 1) 2 l i n e a r l y i n d e p e n d e n t
3
Y~.
T h e s e Y~ i n p o l a r c o - o r d i n a t e s (r, 01, 0~, ~0) of E 4 are p o l y n o m i a l s i n sines a n d cosines of 01, 0~, q. As i t is k n o w n , in z o n a l c o - o r d i n a t e s Z1 =
r,g 1 ,
where x-l--x
rx 2 ,
z2 :
2 l ~ x , ~20 ,
z
-
-
r ~ / x cos q ,
z4 = r ~ / x sin ~v,
z5 = t ,
one c a n t a k e as Y , t h e f u n c t i o n s e x p [ 4 - i~,~1' P("k) (-~.~p '~x 1, x2),
w h e r e t h e p(.k) (0 < k < rt) a r e defined b y
p(,~) r
. (xl ' x2) ~
x ki~ e~V,,,,(Xl, x~) ~x~l ~x~,
( k l + k2 = k ~ , \.rq + ,n2 = ,~
/
(t) M. FLATO, D. STERNIIEL',IER and J. P. VI(~mR: Compt. Rend., 260, 3869 0965). (~) M. FLhTO and J. STERNHEL~,~.ER: C o ? n ~ o t . Rend., 259, 3455 (1964). (s) G. FUBINI: Annali di .~llatemalica (3), 8, 39 (1903).
ON T t [ ~
)lASSES
OF
NONSTRA-N'GE P S E L ' I ) O S C A L A R 3I:ESONN ]~TC.
433
w h e r e tile V~,,.(x~, x 2) a r e t h e p o l y n o m i a l s defined, e.g. b y HERM1TE iS), b y t h e d e v e l o p m e n t of t h e g e n e r a t i n g f u n c t i o n ~
a. 1 a 2 ~ " ~ ( x
l , x.~) .
If we n o w look for t h e levels - - m ~ i n t h e H i l b e r t s p a c e g e n e r a t e d b y t h e f u n c t i o n s ~, = e i~t Y,, w i t h ilie t o p o l o g y of L"(S 3 x I l ) , w e g e t m~ + ( 1 / r " ) n ( u :- 2) -~ m,~. T h e s e are, t h e r e f o r e , t h e e i g e n v a l u e s of o u r s t a t i c s o l u t i o n . M a t h e m a t i c a l l y we m a y c o n s i d e r t h e V4 ~ H a • R, w h e r e H a is t h e single s h e e t h v p e r b o l o i d .z ~2 ~3 - - ~ 4 - r ( e m b e d d e d in t h e p s e u d o - F . u c l i d e a n s p a c e E ~ i s o m e t r i c a l l y ) , a n d we c a n m a k e t h e s a m e a r g u m e n t ((, s u b s t i t u t i n g ~ i n t o i~0 ~). I n t h i s case t h e g r o u p of m o t i o n will b e O(3,1) x R. I n o r d e r to h a v e f u n c t i o n s w h i c h v a n i s h w h e n ~ - > -~ (~ (i.e. a t infinity), we take the 89 1) f u n c t i o n s (for tixed ,~) e-~r ,~",~~ ( 0 < k < n ) , and then consider t h e H i l b e r t space g e n e r a t e d b y t h e f u n c t i o n s :
C o n t i n a i n g in t h e s a m e m a n n e r a.s before, we o b t ~ i n for t h e e i g e n v a l u e s --m~] of t h e L a p l ~ e - B e l t r a m i o p e r a t o r t h e s a m e f o r m u l a m,2 - : ~ 0~ r ( 1 / r 2 ) n ( n - 2 ) . N'otice t h a t in b o t h eases, in t h e l i m i t of fiat s p a c e ( r ~ oo), we g e t t h e basic level ,m~ : m 2. T h e i n t r o d u c t i o n of c u r v a t u r e in s p a c e r e m o v e s t h e ~, degeneracy,> b y g i v i n g rise to s u p p l e m e n t a r y levels. X o t i e e also t h a t in b o t h cases for a g i v e n ~t, l h e r e is o n l y o n e f u n c t i o n a m o n g t h e ( ~ - i 1) "~ (or t h e ~n(,rb *-1) i n t h e s e c o n d case), w h i c h is o n l y a f u n c t i o n of t h e d i s t a n c e in t h e c u r v e d space. ( F o r i n s t a n c e in t h e tirst case t h i s is *he f u n c t i o n (sin(,r~ t - l ) o / r ) / ( ( n + 1 ) s i n L~/r), w h e r e e is t h e d i s t a n c e on t h e t h r e e - d i m e n s i o n a l s p h e r e of r a d i u s r.) I t w a s i n d i c a t e d e m p i r i c a l l y (7) t h a t a mass f o r m u l a c o n t a i n i n g p h e n o m e n o l o g i c a l t e r m of a n i n t e g e r m u l t i p l c of a n ~he p i o n m a s s gives a good d e s c r i p t i o n of t h e m a s s s p e c t r u m of s t r o n g l y i n t e r a c t i n g particles. If we t a k e t h e p a r t i c u l a r choice of p a r a m e t e r s , so t h a t for ~ - 0, m - - v (m,~), a n d t h a t r ~ I/~.~(m~) we o b t a i n m
v(n ~ 1)
in--0,
1,2...).
I n t h i s case. for ~ = 3 we o b t a i n t h e mass of t h e r,-meson, a n d for n = 6 we o b t a i n t h e mass of tile X ~ ( -- "r~-2r:) meson, n a m e l y we o b t a i n b y t h e aid of t h e ~,.l~ q u a n tuna n u m b e r ,> a s e p a r a t i o n b e t w e e n l h e m a s s e s of X o a n d "r,, w h i c h h a v e t h e s a m e JJ', I . I s, ); q u a n t u m n u m b e r s . Of course, in t h i s p a r t i c u l a r case of a m a s s f o r m u l a of t h e k i n d m - Y , % ( N - - 1, 2 .... ), we h a v e to e x p l a i n w h y we find l a r g e g a p s in t h e s p e c t r u m ( a p p a r e n t l y of t h e t y p e ~ ' = 2, 3, 5, 6, 8, 9 .... for n o n s t r a n g e p s e u d o s c a l a r mesons). The a n s w e r m i g h t be g i v e n b y a m o r e d e t a i l e d s t u d y of possible s t r u c t u r e a n d m e c h a n i s m of d e c a y m o d e s of e x i s t i n g p a r t i c l e s of t h i s series. Our l a s t t a s k will be to a n a l y s e wha~ we h a v e d o n e f r o m t h e p h y s i c a l p o i n t of view. F i r s t we choose a g r o u p . f m o v e m e n t of t h e l y p e 0 4 • or O ( 3 , 1 ) x / L
(~) CII. [IERSIITE: Oeltl!r(,~. VO]. 9,, (I)aris, 1908), p. 3 . 9 . (7) 1{. 5I. STEIiNFIEISD.:R : 1~.~1.~'. liet'. L e t t . , 13, 37 (1961), see ~d~o: P h y s .
If, co., 136, :B 1361 (1961).
434
M. FL&TO~ D.~ S T E R N I ~ E I ~ E I ~ ,
J.
STE:RNttrilMEII~
J.
r.
VIG1}~'R
and
G. VtATAGHIN
(Notice that the Lie algebra of the first direct factor appears i n t h e different chains of Lie algebras beginning with that of SO(4,2)--the conformal Lie algebra.) Then we write down the corresponding four-dimensional manifolds. The LaplaceBeltrami operator is then introduced in the local co-ordinates of our manifolds and we assume that its eigenvalue --m~ correspond for different values of n to different masses of pseudoscalar mesons (as a matter of fact, we used our result only for no,strange pseudoscalar mesons as in this case degeneracy is indicated by experiment, and our t r e a tm e n t is only a (( degeneracy-lifting t r e a t m e n t ~), namely we assume the Laplace-Beltrami eigenvalue problem to represent the Klein-Gordon equation ill curved manifold. This may be justitied mathematically, since the L. B. operator is a scalar on the manifold, and as the manifold tends to fiat space the L.B. operator tends to the Klein-Gordon operator. The geometry (for instance in the case of fhe group of movement 04 >', R) is quite simple: in our manifold Sa• R, R stands for the time axis. and S a is our cm'ved description of the usual space in the domain of nuclear forces: due to the plan mass usual sparse is curved, and it is the curvature which plays the role of potential, which is responsible fox" the obtained remus spectrum and which is analogous to the Coulomb potential in atomic physics. This very simple and primitive description might justify ~he numerical choice of ~ = 1/v/~;~> to be the Compton wave length of the plan ( h - - c = 1). ~[ore general models involving at the same time description of internal and external symmetries in curved mauifolds are now examined.
