Volume27B, number 6 P H Y S I C S L E T T E R S 19 August 1968

" M A S S F O R M U L A " F O R M = 0 A N D M = 1 T O L L E R F A M I L I E S

P. DI V E C C H I A and F. DRAGO * L a b o r a t o r i N a z i o n a l i d e l C N E N , F r a s c a t i (Rorna) I ta l y

Received 29 June 1968

A method to derive the so-cal led mass formulae, based on analyticity alone, is presented. The s impler M = 0 case is discussed in some detail. The behaviour of the residue functions is also studied; using the factorization theorem the Tol ler pole is reconstructed. The mass formula for M = 1 is given.

I t i s w e l l known tha t R e g g e p o l e s o c c u r in f a m i l i e s wi th i n t e g r a l spac ing a t z e r o e n e r g y . T h i s c i r - c u m s t a n c e r a i s e s s o m e v e r y i n t e r e s t i n g p r o b l e m s . In f a c t i f the d a u g h t e r t r a j e c t o r i e s a r e a p p r o x i m a t e l y l i n e a r and p a r a l l e l to the p a r e n t fo r p o s i t i v e t, they could g ive r i s e to p a r t i c l e o r r e s o n a n c e s wi th we l l de f ined quan tum n u m b e r s . I t s e e m s t h e r e f o r e i m p o r t a n t to obtain s o m e g e n e r a l i n f o r m a t i o n about the s l o p e s of the d a u g h t e r t r a j e c t o r i e s . S o m e w o r k in th i s d i r e c t i o n has been done by D o m o k o s [1] and D o - m o k o s and Su rany i [2]. F o r s m a l l v i o l a t i o n s of SL(2, C), u s ing the B e t h e - S a l p e t e r equa t ion and a p e r t u r - ba t ion expans ion , they ob ta ined , fo r f a m i l i e s of po l e s with T o l l e r quan tum n u m b e r M = 0, the e x p r e s s i o n

a n ( t ) = a - n + [a 1 +a2 (a - n ) ( a - n + 1)]t + O(t 2)

w h e r e the i n d e x n e n u m e r a t e s the t r a j e c t o r i e s , and, h e r e and in the fo l lowing , a i s the t = 0 i n t e r c e p t of the p a r e n t R e g g e t r a j e c t o r y . In t he B e t h e - S a l p e t e r m o d e l the c o n s t a n t s a 1 and a 2 can be c a l c u l a t e d fo r any g iven k e r n e l .

In th i s l e t t e r we wi l l g i v e s o m e r e s u l t s and the g e n e r a l i dea of a me thod , b a s e d only on ana ly t i c i t y , fo r s tudying the n - d e p e n d e n c e of the r e s i d u e func t ions and of the d e r i v a t i v e s of t he t r a j e c t o r i e s a n ( t ) at t = 0 .

F o r M = 0 the m o s t g e n e r a l n - d e p e n d e n c e of the d e r i v a t i v e s , c o m p a t i b l e wi th ana ly t i c i t y , i s in a g r e e - m e n t wi th the D o m o k o s m a s s f o r m u l a :

an(0) = a 1 + a 2 ( a - n ) ( a - n + 1) . (1)

We h a v e shown e x p l i c i t e l y tha t one a r r i v e s at th i s r e s u l t s t a r t i n g f r o m both the u n e q u a l - u n e q u a l (UU) and the e q u a l - u n e q u a l (EU) m a s s p r o b l e m s .

We h a v e a l s o d e t e r m i n e d the n - d e p e n d e n c e of the r e s i d u e func t ions , both in the UU and EU ease : o n c e th i s i s done, a s s u m i n g the f a c t o r i z a t i o n t h e o r e m , we ob ta ined the n - d e p e n d e n c e of the r e s i d u e func t ions in the equa l (EE) m a s s ca se : the r e s u l t s a r e in a g r e e m e n t wi th t h o s e found u s ing the 0(4) s y m m e t r y . We h a v e thus f o r m a l l y r e c o n s t r u c t e d the M = 0 T o l l e r po le u s ing a n a l y t i c i t y and f a c t o r i z a - t ion.

Us ing e s s e n t i a l l y t he s a m e m e t h o d we h a v e s tud ied the s a m e p r o b l e m f o r the m o r e i n t e r e s t i n g c a s e of f a m i l i e s wi th M = 1, w h e r e the p a r i t y doubl ing p h e n o m e n o n b e g i n s to a p p e a r . We have found the n - d e p e n d e n c e ~-

a n (0) = c 1 + [c2:~c3]((~ - n ) ( a - n + l ) . (2)

Le t us b r i e f l y i l l u s t r a t e h e r e , in the s i m p l e r c a s e of the s c a t t e r i n g of s p i n l e s s p a r t i c l e s (i .e. f o r c l a s s I po l e s ) , the m e t h o d used . A m o r e c o m p l e t e d e s c r i p t i o n of our w o r k wi l l be pub l i shed e l s e w h e r e .

The con t r i bu t i on of a f a m i l y of R e g g e p o l e s to the s c a t t e r i n g a m p l i t u d e i s g iven by

* Sponsored in part by the Air Force Office of Scientific Research through the European Office of Aerospace Re- search, OAR, United States Air Force , under contract F61052 67 C 0084. The ± sign re fe r s to the natural parity. As before a = a+(0) = a- (0) .

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f ( s , t ) = ~ 2 a n + l n=O s i n ~ a n [1 + r n exp ( - i l r an ) ] ~ a n ( - c o s Ot)fln(t) (3)

w h e r e

tg~a 2 a r ( a +½) z a F ( - ½ a , ~ - a , a - a ; z -2) = ~r(a71) * a * 2 a r ( a +½) z a ~ ak(a)z-2k ~a(z) = - ~ ~ - a - l ( Z ) - 4-~r(a + l) k=O "

In the UU c a s e , wh ich we d i s c u s s h e r e , one has

t ( t - ~ m 2) +(rn~- 2m3)(m 22-rn2) s i

cos 0 t - - - + 2qinqou t 4t q inqou t

w h e r e q in and qou t a r e the i n i t i a l and f ina l m o m e n t a in the c r o s s e d t - c h a n n e l . Expand ing the r i g h t hand s ide of eq. (3) in a p o w e r s e r i e s , a f t e r s o m e r e a r r a n g e m e n t , we ge t

-m N(m) m-2k On(t)+n f ( s , t ) = ~ m s ~ ~ bn;k(t ) (_~o j

m=O k=O n=O

w h e r e s o i s a s c a l e f a c t o r ,

_ _ m3)(m 2 - B(t) = i

2s o

½m ff (-1) m = 1

N(m) = l { ½ ( m - l ) ff (-1) m = - l

and, a t t = 0, which i s the po in t of i n t e r e s t h e r e ,

%~m~(0) = g ~ (a) [2 (a -n) + 1] 2n+2kk:

( - l>n~n(0)

r ( ½ - a +n +k)(m -n - 2k)!

w h e r e

½[1 + r e x p ( - i ~ a ) ] 2 a + l

gin(a) = ~ s in 2~a F (a +1 - m )

and Yn(0) a r e t he r e d u c e d r e s i d u e func t ions , f r o m which , f o r n >/ 1, the a p p r o p r i a t e s i n g u l a r i t i e s [3] a t t = 0 h a v e b e e n r e m o v e d .

If t he a m p l i t u d e h a s to be a n a l y t i c at t = 0 we m u s t r e q u i r e tha t , f o r t --. 0

N(m) m-2k

k=O n=O bn;k(t) \-~o] = o(tm) "

T h e s e a r e the f u n d a m e n t a l equa t ions in ou r a p p r o a c h . F r o m t h e s e cond i t i ons we can e x t r a c t the r e l a - t i ons b e t w e e n the r e s i d u e s of the p a r e n t and d a u g h t e r t r a j e c t o r i e s and, d i f f e r e n t i a t i n g with r e s p e c t to t, the n - d e p e n d e n c e of the d e r i v a t i v e s an (0 ) , i . e . the m a s s f o r m u l a . In f ac t a t t = 0 the s y s t e m (4) t u r n s out to be

m [ 2 ( a - n ) + l ] (-1) n 1 2m+nF(m-a)

2 n 22a+14-~ (m-n) ! F ( m + n - 2 a ) 7n(O) =0 n=O

w h o s e so lu t i ons can be shown to be

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7n(0) = - (2a + 1) (-1)n F(n - 1 - 2 a ) " ( 5 )

Once these re la t ions a r e known we can deduce the mass formula . In fact f rom the sys tem (4), dif- fe ren t ia t ing with r e spec t to t, we obtain, for m >/ 2

N ( m ) m - 2 k m ,

~ bn;k(O)an(O) = 0 i

k=0 n=O

F r o m this sys tem we can express the slopes of al l the daughter t r a j ec to r i e s in t e r m s of those of the pa ren t and of the f i r s t daughter in the form

a~(0)n(1 +2a - n ) a'o(O)(n - 1)(n - 2a) v

an(O) = 2a + 2a

which is readi ly seen to be equivalent to eq. (1). At this point it is i n t e r s t ing to invest igate the EU problem, as a f i r s t step toward the proof of the

cons is tency of the whole scheme. Assuming that the equal mass pa r t i c l e s a re pa r t i c l e - an t ipa r t i c l e sys tems , only the daughters with P = G(-1)I contr ibute: for a c lass I family these a re the even one.

Using essen t ia l ly the same methods desc r ibed above we get

EU (-1) n+l r ( n - ½ - a ) EU ~2n (0) = (2a +1) ~o (0) 2n'. r'(~ -a)

and a~(0)n(1 +2a -2n) a 'o(O)(n- 1)(1 +2n - 2 a )

a'2n(O) = 2a - 1 + 2a - 1

which, as expected, is again of the form of eq. (1). Having de te rmined the mos t s ingular par t of the res idue functions in the EU and UU case, us ing the

fac tor iza t ion theorem [4] we can obtain, at t = 0, the daughter res idue in the EE ease, where, as is well known [5], analyt ic i ty i t se l f does not give any informat ion. In this way we obtain

(2n)'. I ' ( a - n + l ) r ( ~ + a ) vEE(0). (7) EE :¢2n (0) - (n!)222n r ( a + 1) F(a + ~ - n)

F r o m the O(4) s y m m e t r y [5], in the sp in less equal mass case, one has

EE R (a +1) 2 .a ,0 ,1 , 2 ~2n (0) - 2~r 2 [2(a - 2 ~ + 1] ]cta-2n;0;0 t ~ ) I

where R is the Tol le r pole res idue and _1

i ( 2 j + l ) F ( n _ j + l ) ] 2 [ f ~1 F(l+z~-(n+j))

L J

It is easily seen that the residues given by the 0(4) symmetry in the equal mass case coincide with those, given above, der ived f rom the usual Regge pole theory.

We have thus r econs t ruc t ed the contr ibut ion of the Tol le r pole to the sca t te r ing ampli tude, and it is i n t e res t ing to note that this has been done, via fac tor iza t ion, through the study of the unequal mass problem, where the extension of the g roup- theore t i ca l approach is not t r iv ia l .

In conclusion, the main r e su l t s of our work have been to es tabl ish the mass formula , eqs. (1) and (2), both for M = 0 and M = 1, s ta r t ing f rom genera l analyt ic i ty p roper t i e s , without any addit ional dynamical assumpt ion. We have also shown, for the s imp le r M = 0 case, that the group- theore t ica l r e su l t s on the res idue functions can be obtained f rom the analyt ic i ty r e q u i r e m e n t s and the factor izat ion theorem.

We a re pa r t i cu l a r ly grateful to Prof . L. Bertocchi , Dr. B. Tagl ient i and Prof. M. Tol le r for useful d i scuss ions . After the conclusion of this work we l ea rned that the M = 0 case was independently studied by Bronzan, Jones and Kuo.

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References 1. G.Domokos , Phys . Rev. 159 (1967) 1387. 2. G. Domokos and P. Suranyi, Budapest p repr in t s (1968) ;

a) Fam i l i e s of Regge t r a j ec to r i e s at nonvanishing energy. I b) Fami l i e s of Regge t r a j ec to r i e s at nonvanishing energy . II.

3. D . Z . F r e e d m a n and J . M . W a n g , Phys . Rev. 153 (1967) 1597. 4. M.Gel l -Mann, Phys . Rev. Le t t e r s 8 (1962) 263;

V. N.Gribov and I. Y. Pomeranchuk , Phys . Rev. Le t t e r s 8 (1962) 343. 5. D . Z . F r e e d m a n and J . M . W a n g , Phys . Rev. 160 (1967) 1560.

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