A novel application of Regge trajectories

Volume

28B, number 4

PHYSICS

A NOVEL

LETTERS

APPLICATION

OF

REGGE

9 December

1968

TRAJECTORIES

C. LOVELACE CERN, Geneva Received

31 October

1968

We suggest that Dalitz plots for 3?r decays and annihilations are “measurements ” of ~71scattering in the double spectral region. Veneziano’s formula for crossing-symmetric Regge trajectories both justifies the external mass extrapolation and correctly predicts the experimental results. It also fulfils Adler’s self-consistency condition if flP( w$) = 4.

Recently Veneziano [l] gave a simple formula which exhibits the Regge pole = resonance duality [z]. We show here how this new development in Regge theory can be applied in a rather unexpected direction, to explain three-particle final state interactions. These then provide direct experimental evidence that Veneziano’s formula is not a mere model, but is approximately true in nature. There is also reason, from at least seven parallel predictions, to believe it connected with chiral symmetry. To start, consider xIT+v-elastic scattering. There are no u channel resonances, since these would have isospin 2, while the s and t channels are identical. Veneziano’s formula [l] then requires exchange-degeneracy between the fo and p trajectories (in agreement with experiment [3]) and gives for the v+v- scattering amplitude

A(spt) =- p

Ul - 4s)) r(l - iJ (0) l-(1-o!(S)-a(t))

(1)

l-(1 -a(s)) l-0 - a(t)) + . . . +y

r(2-a(S)-a(t))

The vv amplitudes for the three isostates in the s channel will be A0 = f[A(s, t) +A(s,u)] A1 = A(s, t) - A(s,u)

- $A(t,u) ,

, (2)

A2 = A(t, u) . /3 in (1) is a constant, and its coefficient conbesides the p and fo trajectories, unitspaced daughters of alternating isospin. The y term contains the daughters only (and might cancel them), while further terms start with the second daughter. Veneziano’s formula is only strictly valid when the trajectories are exactly linear, so that resonances are approximated by

tains,

264

poles [4]. Now consider what happens when we take one of the external pions off its mass-shell. Compari. son with Feynman diagrams shows that the poles of (1) must be in the variables s = (kl + k2)2, t = (kl- ki)2, i.e., that the Regge trajectories cannot depend kinematically on the external masses. Only p, y etc. can therefore vary as we go off- shell. Adler’s self-consistency condition [ 5] (recall that this is a dynamical constraint, not true for individual Feynman graphs) requires (1) to vanish when s = t = u = ~2 and one of the pions has zero mass. Here ~1is the physical pion mass. The first term of (1) contains a factor Q(S) + o (t) - 1

(3)

which will vanish there provided (u(lL2) = + .

(4)

If we take the p trajectory as linear, and the p mass as 764 MeV, this gives op(O) = 0.483 MeV. Now ap(s) must deviate from linearity because of unitarity corrections, and we can estimate the size of these by comparing different determinations of (~~(0) from Chew-Frautschi plots and Regge fits. Our table shows that (4) is true to within the accuracy to which we expect the Veneziano formula itself to hold. Thus the Veneziano formula with one term only [y, etc. = 0 in eq. (l)] and the experimental trajectory predicts Adler’s self-consistency condition. (We shall denote such parallelisms, when a chiral symmetry prediction is also a Veneziano prediction, by roman numerals. This is I.) Either this is coincidence and the higher terms just happen to cancel at this point, or else there really is only one Veneziano term present in the ~TBcase. We shall assume the latter. The same Regge trajectories will occur wher-

PHYSICS

Volume 28B, number 4 Table 1. Old and new determinations of the

Old

New

)

intercept

Method

~p(O)

Regge fit to charge-exchange [21]

0.57 i 0.01

Finite energy sum rule [3]

0.55

Linear interpolation between the dip and the p

0.485

Linear extrapolation from the p and g

0.462

Soft pion theorem

0.483 0.491 * 0.002

Fit to Tdecay (data of ref. 19)

0.528 * 0.006

e v e r a l l f o u r e x t e r n a l p a r t i c l e s h a v e the q u a n t u m n u m b e r of the pion. If one i s a z e r o m a s s pion, t h e n A d l e r ' s s e l f - c o n s i s t e n c y c o n d i t i o n m u s t ag a i n hold. To get the r e q u i r e d z e r o s f r o m the V e n e z i a n o f o r m u l a , a l l a c t u a l p a r t i c l e s w i t h the pion's quantum numbers must satisfy n = O, 1, 2, . . .

(5)

T h i s i s r e a s o n a b l e , s i n c e the d a u g h t e r s of the p i o n ' s R e g g e r e c u r r e n c e would b e j u s t h e r e f o r parallel trajectories. Adler's self-consistency c o n d i t i o n s t h u s r e q u i r e s q u a n t i z a t i o n of the e x ternal masses, which is strange viewed from conv e n t i o n a l t h e o r y , but by no m e a n s u n p h y s i c a l . If a p a r t i c l e s u c h a s 77, w h o s e m a s s d o e s not c o i n c i d e w i t h a p i o n r e c u r r e n c e , d e c a y s into t h r e e pions through electromagnetic or weak interact i o n s , t h e n V e n e z i a n o ' s f o r m u l a p r e d i c t s that only one of the t h r e e s o f t pion z e r o s [6] w i l l o c c u r . This is experimentally correct, as we shall see. T h e l i n e a r t r a j e c t o r y w h i c h g i v e s eq. (4) and the p m a s s is a (s) = 0.483 + 0.885 s

(6)

If we now e x p a n d the f i r s t t e r m of eq. (1) a s a s u m of p o l e s A ( s , t) = #[ol (s) + ~ (t) - 1] x

×

(7)

n n=0

l ) (n

n

and d e c o m p o s e the r e s i d u e s into p a r t i a l w a v e s , we o b t a i n the r e l a t i v e 7nr w i d t h s of the v a r i o u s daughters: p daughter widths

P 1 : SO = 2 : 9

fOdaugther widths

DO : P 1

9 December 1968

[ T h e r o u n d n u m b e r s would be e x a c t if ~ = 0. ] C h i r a l s y m m e t r y a l s o p r e d i c t s [7] e q u a l i t y of the p and e m a s s e s (II) and the width r e l a t i o n (8a) (III). The l a t t e r i s a V e n e z i a n o p r e d i c t i o n only if y = 0 in eq. (1). S u b s t i t u t i n g eq. (6) and the f i r s t t e r m of eq. (1) into eq. (2) g i v e s the 7r~ s c a t t e r lag lengths a0 = 0.395fl,

Fit to 7/decay (data of ref. 17)

q p ( m 2) = ~1 + n ,

LETTERS

(8a)

: S 0 = 9 : 10 : 0 (8b)

a 2 = - 0.103 f i .

(9)

The r a t i o i s within 10% of W e i n b e r g ' s v a l u e [8] (IV). To p r o c e e d f u r t h e r we m u s t g i v e the r e s o n a n c e s f i n i t e w i d t h by a d d i n g an i m a g i n a r y p a r t to ~ (s) f o r s > 4p 2. U n f o r t u n a t e l y t h e r e i s no c o n s i s t e n t w a y of d o i n g t h i s . T h e r e s o n a n c e s at the s a m e m a s s w i l l a l l a c q u i r e the s a m e t o t a l width by eq. (7), but m u s t h a v e d i f f e r e n t e l a s t i c w i d t h s by eq. (8), w h i c h c o n t r a d i c t s e l a s t i c u n i t a r i t y . A l s o the b o u n d a r y of the d o u b l e s p e c t r a l r e g i o n w i l l b e at s = 4 ~ 2 , t = 4~2 i n s t e a d of the c o r r e c t c u r v e , so that p a r a l l e l t r a j e c t o r i e s at t h r e s h o l d v i o l a t e the c e n t r i f u g a l b a r r i e r [9]. F u r t h e r m o r e if ~ (t) i s not a p o l y n o m i a l , eq. (7) s h o w s that the s c h a n nel r e s o n a n c e s w i l l h a v e h i g h - spin " a n c e s t o r s " a s w e l l a s d a u g h t e r s . F o r the p only, t h i s l a s t d i f f i c u l t y c a n be a v o i d e d by not g i v i n g any i m a g i n a r y p a r t to the f i r s t z e r o , i . e . , r e p l a c i n g eq. (1) by A(s,t)

= [ 0 . 8 8 5 ( s , t ) - 0.034] x

(10)

r ( 1 - ~ (s)) r ( i - ~ (t)) x~

r(2-a(s)-a(t))

T o a p p l y V e n e z i a n o ' s m a r v e l l o u s f o r m u l a to e x p e r i m e n t , we m u s t t h e r e f o r e c l a i m p o e t i c l i c e n c e to a d j u s t I m a to the n e e d s of the m o m e n t . F o r o u r f i r s t a p p l i c a t i o n the e width w i l l be m o r e i m p o r t a n t t h a n the p, so we t a k e a(s)=0.483

+0.885s

+i0.28~s-

4~ 2 (11)

w h i c h , w h e n eq. (10) i s a n a l y s e d into p a r t i a l w a v e s , c o r r e s p o n d s to an e 280 MeV b r o a d . E x p e r i m e n t a l l y [10] it s e e m s to be b e t w e e n 250 and 500 MeV. Now we c o n t i n u e one of the e x t e r n a l p i o n s to a l a r g e p o s i t i v e m a s s , i . e . , we c o n s i d e r the 37r d e c a y of a p a r t i c l e w i t h the s a m e q u a n t u m n u m b e r s a s t h e pion. T h i s w i l l h a v e the s a m e R e g g e t r a j e c t o r i e s a s 7rTr s c a t t e r i n g so the V e n e z i a n o f o r m u l a can only d i f f e r in the v a l u e s of /3, y, etc. in eq. (1). With a s i n g l e V e n e z i a n o t e r m , we can t h e n p r e d i c t e v e r y t h i n g e x c e p t the t o t a l d e c a y r a t e . The D a l i t z p l o t d e n s i t i e s f o r v a r i o u s c h a r g e c o m b i n a t i o n s s h o u l d be [X i --* ~r+Tr+TrT] = N Id (s, t )l 2 ,

(12a) 265

Volume28B, number 4 iX + ~ ~±~ono]

=

PHYSICS LETTERS

iX o _~ 7r+TT-no]

=

(12b)

9December 1968

3.0 • "6~. : "-~'E. .e...

= } N I ~ (t, u) - A ( s , t) - A ( s , u)] 2 ,

Q)

2.5

(12c) iX ° ~ 37r° ] =~ ¼NIA (s,t ) + A ( s , u) + A(t, u ) ] 2 . H e r e N i s a n o r m a l i z a t i o n constant, s, t, u a r e the s q u a r e s of the m a s s e s of the final dipion c o m b i n a t i o n s ( s y m m e t r y shows which). If the decay p r o c e s s couples to all yT~ r e s o n a n c e s , the A ( s , t) i s given by eqs. (10) and (11). Thus, if the Veneziano f o r m u l a r e a l y d e s c r i b e s the i n t e r f e r e n c e of s and t channel Regge t r a j e c t o r i e s , then it m u s t n e c e s s a r i l y a l s o d e s c r i b e how o v e r l a p p i n g r e s o n a n c e s i n t e r f e r e in t h r e e - p a r t i c l e decays. T r a d i t i o n a l l y , o v e r l a p p i n g r e s o n a n c e s in i s o b a r m o d e l s have s i m p l y been added in the m a t r i x e l e m e n t , though the c o r r e c t n e s s of this has b e e n doubted [11]. According to the Veneziaao f o r m u l a , duality should occur, e x p r e s s e d by the equality of the s and t channel f o r m s of eq. (7). We shall now test this e x p e r i m e n t a l l y . In addition to the c l a s s i c K ~ 3~ and 7 / ~ 3y d e c a y s , it is i n t e r e s t i n g to c o n s i d e r pn ~ ~+~r-nat r e s t , b e c a u s e the i n i t i a l state is known e x p e r i m e n t a l l y [12] to be e n t i r e l y 1S0, a s we might expect at threshold. This has the q u a n t u m n u m b e r s of the pion. No p is o b s e r v e d in the 3n Dalitz plot [12]. We t h e r e f o r e have to r e m o v e the l e a d i n g t r a j e c t o r y f r o m the Veneziano f o r m u l a by c r o s s ing out the f i r s t factor of eq. (10). T h i s decoupling of the p is a p u r e l y phenomenological a s s u m p tion, but any theory of final state i n t e r a c t i o n s applied to this decay i s forced to make it, a s t h e r e is nothing in the final state to forbid p. Taking a s i n g l e Veneziano t e r m in eq. (12a), we then have a 1½ p a r a m e t e r fit, since the n o r m a l i z a t i o n is not p r e d i c t e d and the • width is only roughly known f r o m other e x p e r i m e n t s . The 0.28 in eq. (11) was in fact obtained f r o m this fit. The r e s u l t i n g Dalitz plot i s shown in fig. 1. The a g r e e m e n t would p r o b a b l y be still b e t t e r with a m o r e c o m p l i c a t e d Im a (s) to make the • b r o a d e r below than above (in a c c o r d with other evidence [10]), but it i s a l ready r e m a r k a b l y good for so few p a r a m e t e r s . T h i s s t r a n g e Dalitz plot i s just what to expect f r o m the Veneziano f o r m u l a . The c r i s s - c r o s s of v e r t i c a l and h o r i z o n t a l poles, with diagonal z e r o s to c a n c e l the double poles at t h e i r i n t e r s e c t i o n s [4], i s c l e a r l y v i s i b l e . The hole in the middle c o m e s f r o m the zero at ~ (s) + ot (t) = 3, and thus c o n f i r m s that this i s not filled i n by s e c o n d a r y V e n e z i a n o t e r m s . Even m o r e s p e c t a c u l a r i s the e n h a n c e m e n t at the upper right edge. Any naive i s o b a r model m u s t a t t r i b u t e this [13] to a s t r o n g 266

A 1.5

'~.°.'<":~::--

" . 2 " "£ "-

~,, 1.0

0.G

s=~lZ(w*~ -)

(GeVlc2) z

N

b) 2.10(

0

0.5

tO m2.n -"

1.5

ZO

Z5

3.0

20C N

pn ~

(It-lt-)It"

c)

In0

05

1.0

1.5

20

2.5

30

-- rn ~-1c

Fig. 1. Dalitz plot and its projections for ~n ---,n--a--~~at rest (data of ref. 12). The solid curves and contours are from the Veneziano formula with the trajectory (11)~ the dashed curve is phase space. low e n e r g y 7U~- i n t e r a c t i o n in the i s o s p i n 2 state - a m o s t unpalatable hypothesis. H e r e it c o r r e s ponds to the r e g i o n w h e r e s and t a r e both l a r g e and positive, and d e m o n s t r a t e s the blow-up of the double s p e c t r a l function in this d i r e c t i o n , for a t h e o r y with r i s i n g Regge t r a j e c t o r i e s A c c o r d ing to the Veneziano f o r m u l a , this Dalitz plot is a window through which we can o b s e r v e 7rTrs c a t t e r i n g in the double s p e c t r a l region. F u r t h e r -

Volume28B, mtmber 4

PHYSICS LETTERS

m o r e , a s the p d e c o u p l e s , V e n e z i a n o ' s f o r m u l a s a y s the fo will a l s o de c o u p le , so the second bump in fig. l b m u s t be the long sought p ' . Since eq. (11) e v i d e n t l y f i t s , we can u s e it to d e t e r m i n e the p ' m a s s and width a s 1310 and 400 MeV, which s e e m r e a s o n a b l e v a l u e s f o r the e l e c t r o m a g n e t i c f o r m

NILII 2.(;

//'S

i

(:15

'

~1o

'

o'1~..2._ Q~

o'16

N/,.p

i

1.3

K'-~ (~-"~'- )'It;

Q? 008

2.(

00~

OIC m2

K° ~(~* It-)~ °

011

0'12 -~l.

K°--0~o~o}~ *

T• ~

"~ ~t l

Fig. 2. Spectra (divided by phase space) for the bracketed ~'~ combinations in various 7/ and K decays (data of refs. 17-19). The solid curves are zero-parameter predictions using eq. (11), the dashed curves are one-parameter fits relaxing the soft pion constraint (see table 1).

9 December 1968

f a c t o r s [14]. Now we c o n s i d e r ~ and T decay, which b e h a v e a c c o r d i n g to p r e v a l e n t t h e o r i e s [e.g. 15] a s if the i n i t i al p a r t i c l e had the quantum n u m b e r s of the pion. H e r e we do not of c o u r s e decouple the p, si n ce the f i r s t f a c t o r of eq. (10) will a u t o m a t i cal l y give the z e r o r e q u i r e d by soft pion t h e o r y f o r T decay [16] (V), and one of i t s z e r o s [6] for 77 d ecay (VI). As we explained above [eq. (5)] t h e r e is only one z e r o b e c a u s e the T/ m a s s does not coincide with a pion r e c u r r e n c e . No a r b i t r a r y a s s u m p t i o n about l i n e a r m a t r i x e l e m e n t s is now needed. V a r i o u s nTr s p e c t r a divided by p h ase space a r e shown in fig. 2. (We did not a t t e m p t pion m a s s - d i f f e r e n c e c o r r e c t i o n s . ) The solid c u r v e s a r e z e r o - p a r a m e t e r p r e d i c t i o n s (since the total decay r a t e does not en t er ) , obtained by substituting eqs. (10) and (11) into eqs. (12a) and (12b). F o r ~7 decay [17] the a g r e e m e n t i s obviously ex c e l l e n t , but for T decay though m o s t of the ex p e r i m e n t s [18] fit, the l a r g e s t [19] if taken l i t e r ally (this is p r e l i m i n a r y data u n c o r r e c t e d f o r systematic e r r o r s ) suggests some discrepancy. The dashed l i n e s in fig. 2 a r e o n e - p a r a m e t e r fits in which the soft pion c o n s t r a i n t eq. (4) was r e laxed. The r e s u l t i n g ap(0) a r e shown in table 1. We see that the s p r e a d in the new d e t e r m i n a t i o n s i s l e s s than that in the old ones. It s e e m s to us that t h e s e e x p e r i m e n t a l s u c c e s s e s a r e a good r e a s o n f o r taking the V e n e z i a n o f o r m u l a s e r i o u s l y , not as a model, but as an ap p r o x i m a t e t h e o r y of the s t r o n g i n t e r a c t i o n s . The quantity of p r e v i o u s p a p e r s on the su b j ect shows how n o n - t r i v i a l it is to account for the T and 77 d ecay final state i n t e r a c t i o n s without conflicting with o t h e r v~ ev i d en ce. U n l e s s we b e l i e v e in a low m a s s i s o s p i n 2n~ r e s o n a n c e or v i r t u a l bound state, ~n ~ 3n has no o t h er explanation at all. With a p p r o p r i a t e t r a j e c t o r i e s our t h e o r y can be applied w h e r e v e r t h r e e p a r t i c l e s o c c u r with d efinite total spin. We e m p h a s i z e that I m a (s) should be chosen in each p r o b l e m to give the widths of the p a r t i c u l a r c o n t r i b u t i n g r e s o n a n c e s - t h e r e can be no u n i v e r s a l l y v al i d f o r m u l a f o r it. P e r h a p s the m o s t exciting p r o s p e c t , h o w e v e r , is of unifying Regge t h e o r y and c u r r e n t a l g e b r a . M o r e p a r a l l e l i s m s and c o n v e r g e n t r e s u l t s can e a s i l y be d i s c o v e r e d . Thus, si n ce the V e n e z i a n o f o r m u l a e x t r a p o l a t e s c o r r e c t l y f r o m mn to 2raN, it should convey us safely f r o m 0 to m K. We then get for all soft m e s o n t h e o r e m s to hold in nK s c a t t e r i n g , the obvious SU(3) ex t en si o n of eq. (4) a K , ( m 2) = ½ ,

(13)

p r e d i c t i n g for p a r a l l e l t r a j e c t o r i e s

267

Volume 28B, number 4

P H Y SIC S L E T T E R S

mi:

m

-

(14)

w h i c h i s a n SU(6) r e s u l t a n d t r u e , t h o u g h t h e d e r i v a t i o n a c t u a l l y c o n t r a d i c t s SU(6) b e c a u s e eq. (13) d o e s not allow v e c t o r and p s e u d o s c a l a r m e s o n s to b e d e g e n e r a t e . A g a i n , if t h e p i o n c o n s p i r a t o r t r a j e c t o r y i s p a r a l l e l to t h e p, eq. (4) g i v e s

mL-

-

which is e x p e r i m e n t a l l y exact and a g r e e s for m y = 0 w i t h a d e d u c t i o n b y W e i n b e r g [20] f r o m a l g e b r a of f i e l d s (VII). W e h a v e now c o u n t e d seven such coincident predictions, which leads to c o n j e c t u r e : Chiral symmetry for soft mesons + absence e x o t i c r e s o n a n c e s = V e n e z i a n o f o r m u l a w i t h no secondary terms. The i m p o r t a n c e of v e r i f y i n g t h i s c a n h a r d l y o v e r e s t i m a t e d - it w o u l d v i r t u a l l y a m o u n t t o a c o m p l e t e t h e o r y of t h e s t r o n g i n t e r a c t i o n s .

11. 12.

13. the 14. us of

15.

be 16.

I a m g r a t e f u l t o D r s . C. S c h m i d , M. J a c o b , R. G. R o b e r t s a n d F . W a g n e r f o r d i s c u s s i o n s .

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268

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9 D e c e m b e r 1968

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