Kinematical enhancements in the final state of peripheral processes

Volume 15, number 3

PHYSICS L E T T E R S

Eq. (17) for the luminc~tty L follows directly from eqs. (6) and (7) and the definition of L. Fteq. (11) for DtUIS just the RO0 Einstein equation (which by e p ~ r i c a l syr~_metry is the same. as the. R ~ o equation), The R ~ equation, which g o v e r n s ' y ( r , t ) , i s a n identity since we have eliminated ~, f r o m the problem using the T ° equations [12], ~One will note in the energy balance equation (13), that the heat input to a sheU of matter is pro~z~tional to the difference in the values of L e °~ at the i n n e r and outer s u r f a c e s of the sheU. Thus, ff no heat is absorbed or given off, the quantity L e ~I~ is constant a c r o s s the shell, and L will d e c r e a s e corresponding to the increase in e ~k . One factor of e @ r e p r e s e n t s the energy r e d shift while the second r e p r e s e n t s the time dilation of the interval during which this radiation is t r a n s f e r r e d . The a r r a n g e m e n t of the above s y s t e m of equations, and in particular the distinction between I" and ~, follows the system which May and White [14] have found suitable for numerical integration in the case without heat transfer. In the absence of nuclear energy generation the above set of equations, together with appropriate boundary conditions ~ ] , completely define the problem. To include nuclear energy generation one must supplement these equations with equations governing the rate of change of the chemical composition. Eq. (13) is still correct, provided one understands the average internal energy per nucleon, e, to include its nuclear binding energy [12].

1 April 1965

The authors wish to thank Mr. J. M. Bardeen and Dr. R. W. Lindqulst for a number of very interesting discussions regarding this problem. 1. C. Eckart, ]Phys. Rev. 58 (1940) 919. 2. L.D.Landmt and E.M.Lifshitz, Fluid mechantc~ (Pergamon Press Ltd., London, 1959) § 127. 3. G.E. Tauber and J. W. Weinberg, Phys. Rev. 122 (1961) 1342. 4. N.A. Cherntkov, Acta Physica Polonica 23 (1963) 629. 5. Such a study has been initiated by J. M. Bsrdeen, by Tauber and Weinberg and by R. W,. Lindqulst. We thank these workers for informing us of their results, and for a number of interesting coLversations. 6. L.H. Thomas, Quart. J. Math. (Oxford) 1 (1930) 239. We thank J. W. Weinberg for calling our attention to this important paper. 7. R.F. Christy, Rev. Mod. Phys. 36 (1964) 555. 8. S.A.Colgate and R.H.White, UCRL-7777 (Rev. Mod. Phys. to be published). 9. S. A. Colgate and R.H. White, Proc. Second Texas Syrup.on Relativistic astrophysics, 1964 (to be published). 10. C.W.Misner and D.H.Sharp, Phys.Rev. 136 (1964) B571. 11. C.W.Misner and D.H.Sharp, Proc.Second Texas Syrup. on Relativistic astrophysics, 1964 (to be published), 12. For a cor,~parisov of the Eckart and the LandauLifshitz [2] expressions for q~ in the diffusion approximation see: C. W. Misner and D. H. Sharp (to be published). 13. D.A. Frank-Kamenetskii, Physical processes in stellar interiors, Israel Program for Scientific Translations, Jerusalem, 1962). 14. R.H. White and M. May, Proc. Second Texas Syrup. on Relativistic astrophysics, 1964 (to be published).

*****

KINEMATICAL OF

ENHANCEMENTS IN T H E F I N A L PERIPHERAL PROCESSES *

STATE

U. MAOR and T. A. 01HALLORAN Jr.

Received 1 March 1965

It has been recently suggested by Deck [1] that the observed [2] lr-p peak A 1 (1085 MeV) in the reaction

* Supported in part by the U.S. Atomic Energy Commission.

+ + p --.~ + + p +

#

(1)

might be understood in terms of simple kinem ~ i c a l features of the one pion exchange diagram (fig. la). According to b_ls analysis, a 281

Volum~ 15. n u m t ~ r 3

l,m,.

PHYSICS LETTERS

s,+ ~...~ '~

O

X

)

to)

(hi

['l~. I. s) O~e pkm exchange for ~'++p-.p°+v+ +p. b~ Perlpberal diagram for A + B -,C + D + E.

str~

enhancement in the mass spectrum of the

final T-o s y s t e m m a n i f e s t s itself in the r e g i o n of the A I peak ff one a s s u m e s that the e x c h a n g e d e.xh/b/ts diffraction s c a t t e r i n g on the t a r g e t proton. However, it is c r u c i a l to note that m o s t ~+-p p a i r s in r e a c t i o n (I) fall e x p e r i m e n t a l l y below the diffraction region. This fact is also r e f l e c t e d in D~ck's calculation of the total c r o s s s e c t i o n for this reaction, which is about 30 t i m e s s m a l l e r than the e x p e r i m e n t a l ~-alue. In this l e t t e r we would like to c l a r i f y these difficulties, and a l s o d e m o n s t r a t e that s o m e of the newly r e p o r t e d r e s o n a n c e s c a n be u n d e r s t o o d klnematically. A s s u m e the r e a c t i o n A+ B--'C+D+E

(2)

is dominated by a one pm'ticle exchange m e c h a m s m (fig. lb}. Our main o b s e r v a t i o n is that a strong f o r w a r d - b a c k w a x d a s y m m e t r y in the c,m, a n . ~ l a r d i s t r i b u t i o n of X + A ~ C + D will be r e [.ec~.d by azl a p p r e c i a b l e e n h a n c e m e n t in the ::-ass s p e c t r u m of E + D near the lower edge of ;...ca" p~.a.se-space distrioution. The m a i n f e a t u r e s of this e n h a u c e m e n t a r e d i s c u s s e d below. Let us r e c o n s i d e r the one pion exchange contribution to r e a c t i o n (1). Following Deck, we evaluate the f o r m f a c t o r s and the 0 spin p r o j e c tioas at the pole x2 = _m ~r 2 and r e p l a c e the +r-p matrkx element ~ , '.If~p 2 = (8~TW)2 d%p/d~2

(3)

~ettn~g: 2 dc : ( ~ _1 ) 2 /-~,'~r=) -,-:;

1

(m - 4 , 4 )

1 April 1985

GeV2), we have followed the A I experlmenta]L a n a l y s i s [2] which e x c l u d e d only those e v e n t s with v a l u e s of ~0 in the (3, 3) N* r e s o n a n c e band. ~as e r t l n g the e x p e r h n e n t a l values of the d i f f e r c n t l a l w+-p e l a s t i c c r o s s s e c t i o n s [4] in eq. (4) y i e l ~ a s t r o n g e n h a n c e m e n t of the final # - p s y s l ~ m in t h e A 1 r e g i o n . This r e f l e c t s the s t r o n g a s y m m e t r y c h a r a c t e r i s i n g the a n g u l a r d i s t r i b u t i o n Of ~+..p s c a t t e r i n g in t h e i r r e s t s y s t e m above the (3, :$) e n e r g y region. Fig. 2 a r e p r e s e n t a the c a l c u l a t e d m a s s s p e c t r u m of u 3, c o r r e s p o n d i n g to an Ir.cident pion m o m e n t u m of $.65 GeV/c, c o m p a r e d with the a p p r o p r i a t e n o r m a l i z e d p h a s e - s p a c e d i s t r i b u t i o n . The graph r e p r e s e n t s a ~-O peak ~Lt 1050 MeV with a h a l f - w i d t h of ~ 50 MeV. The s a l i e n t f e a t u r e of o u r calculation is that the one pion exchange c o n t r i b u t i o n accounts f o r a total

c r o s s section of 0.22 rob. Past experience showed [5] that the approximations used in ttds calculation r e p r e s e n t the shape of the d i f f e r e n t i a l c r o s s s~ctions r a t h e r well, but underesfln...i:e the total c r o s s s e c t i o n by a f a c t o r of 2 to $,, It s c v m s , t h e r e f o r e , r e a s o n a b l e t h a t the one pion e x c h a n g e d i a g r a m (fig. la) p r o d u c e s k i n e m a t i c a l e n h a n c e m e n t of the final ~ - p s y s t e m , which w~3 i n t e r p r e t e d as a r e s o n a n c e . Since the final s t a t e of r e a c t i o n (1) h a s two i d e n t i c a l bosons, a n o t h e r m e c h a n i s m which could p r o d u c e an e n h a n c e m e n t in the final s t a t e is the i n t e r f e r e n c e t e r m b e t w e e n the two d i a g r a m s which have the positive pions i n t e r c h a n g e d . Tiffs cadculation is much m o r e complicated, and c a l l s for s o m e specific a s s u m p t i o n s about the p h a s e s of the r e l e v a n t a m p l i t u d e s . However, it is i m p o r t a n t to notice that the e x p e r i m e n t a l ratio [2] oJ ~r;~r" and 7r~Ir', with both p a i r s falling in the r h o m a s s re-gion, is a p p r o x i m a t e l y the one p r e d i c t e d by p h a s e s p a c e calculations so t h a t one is r a t h e r e n c o u r a g e d to ignnre the i n t e r f e r e n c e t e r m and c o n s i d e r the p u r e amplitude only. In any case, i~ o r d e r to c o m p l e t e l y avoid this s o r t of c o n s i d e r a t i o n , let us e x a m i n e the one pion exchange c o n t r i b u t i o n to

K+ + p - . T r + + p + K';'o |

w2 do~.~

(4) d3old3q2d3 q 54 C-~ + P2- q l - q 2 - q) .................... •

qloq2oqo

We have used the notation introduced by Selleri mad F e r r a x i [3] w h e r e W2 ~2_(p1 + P2)2,2 .~2 = (q2-79)2, t2 : ( q l - P l ) 2 , u = - ( q + q 2 ) , w2 = _ (q+ @2. (s) "~'~<.+:ea~ D e c k ' s calculation [1] was limited to •.a!ue~ in /he =+-p diffraction r e g i o n (w 2 > 2.7

(a)

at 3.0 GeV/c. Our c a l c u l a t i o n yields a s t r o n g enh a n c e m e n t of the 7r+-K *° s y s t e m at 1180 MeV (fig. 2b) with a total c r o s s section of 0.2 mb, If one c o n s i d e r s the fact that K..p s c a t t e r i n g at~ove s e v e r a l hundred MeV is dominated by high p ~rtial w a v e s , and a s s m n e s that the s a m e holds f o r ~-o + p _. it+ + Ao (~o), a s i m i l a r c a l c u l a t e d peak of the 1r-K* s y s t e m s in the final state of

_ + p - + + , _ +.+ + + K o + Ao

Volume 15, number 3

i~)

" 'i '

PHYSICS

~ (~!(,~)

1 ~Z'II 1965

"

(115

0,15

LETTERS

p,

tp

.~ ~ - - q

P _ - ~ ~ - q ,

Fig. 3. p exchange dlagrmn for ~'++p--. ~P+7++p.

IIi

1.0 "'

\

2,0 u' tOeW)

",

5.0

I

I.O

2.0 .I I ! ~ )

3.0

Fig. 2. I1) ~+mO° calculated mass distrtlmtion for reaction (1) at 3.65 GeV/c incident momentum, b) ~+oK*° calculated mass distribution for reaction (6) at 3.0 GeV/c. Both distributions are compared with the appropriate normalized phase-space distributions (dashed line). is obtained. T h i s peak a g r e e s well with the r e ported r e s o n a n c e [6]. S i m i l a r b e h a v t o u r was a l s o e x a m i n e d for ~) exchange d i a g r a m s , e s p e c i a l l y in connection with the B resona~, ~e [7] (see l,~ig. 3). F o r an incident pion m o m e n t u m of 3.5 GeV/c we g e t a pronounced peak at 1200 MeV. In this c a l c u l a t i o n we have r e placed the dl1"ferentlal c r o s s s e c t i o n at the lower v e r t e x of fig. 3 by the e x p e r i m e n t v l d i s t r i b u t i o n s [8] m e a s u r e d in ~r+p --~ p÷p r e a c t i o n s . However, since our a p p r o x i m a t i o n s , i.e., n e g l e c t of f o r m f a c t o r s , the co° spin proJ,ectlons, and other r e l e vant d i a g r a m s and i n t e r f e r e n c e s , a r e certainly questionable, we feel t l ~ t a m o r e c a r e f u l e x a m i nation of this p r o b l e m is worthwhile. ~ x o f e a t u r e s p e c u l i a r to the s u g g e s t e d kinem a t i c a l e n h a n c e m e n t should be noticed: a. The p o s i t i o n of thelm p e a k s a r e energy dependent. At h i g h e r p r i m a r y e n e r g i e s the peaks n a r r o w and shift to lower v ~ u e s of the i n v a r t a n t m a s s . If one a s s u m e s thltt the one pion exchange contribution is s t i l l a p p r e c i a b l e at higher e n e r ~ e s , the position of the #-D peak in r e a c t i o n (1) will move f r o m 1050 MeV at 3.65 G e V / c incident m o m e n t u m to 1020 MeV at 18 GeV/c. As for r e a c t i o n (6), the shift is m o r e a p p r e c i a b l e . At 3 GeV/c we obtain the p e a k at 1180 MeV, and at 18 GeV/c the peak is s h i f t e d to 1100 MeV. A c a r e f u l examination of the e n e r g y dependence of the newly r e p o r t e d r e s o n a n c e s is thus very d e s i r a b l e in o r d e r to e l i m i n a t e m i s i n t e r p r e t a t i o n s of k i n e m a t i c a l effects. b. The c a l c u l a t e d peaks do not fall in delinite

a n g u l a r o r i s o t o p i c spin s t a t e s . Consequently, the b r a n c h i n g r a t i o s and angular d i s t r i b u t i o n s of the involved p a r t i c l e s depend on the s p e c i f i c I n t e r a c tion where s u c h an e n h a n c e m e n t is o b s e r v e d . Det e r m i n a t i o n s of s p i n and i s o s p i n of t h e s e " r e s o n s n c e s " which a r e b a s e d on d i f f e r e n t r e a c t i o n s cotfld, t h e r e f o r e , l e a d to c o n t r a d i c t o r y r e s u l t s . In conclusion we would like to s t r e s s again the n e c e s s i t y of a v e r y careful a n a l y s i s of the newly r e p o r t e d g e n e r a t i o n of r e s o n a n c e s in o r d e r to avoid confusion between k i n e m a t i c a l and dyn a m i c a l f e a t u r e s of the p r o c e s s e s u n d e r d i s c u s sion. Since the e f f e c t s d i s c u s s e d in t h i s le~ter would not be e x p e c t e d to occur in p - ~ annihilation, it would be d e s i r a b l e to verify the e x i s t e n c e of these r e s o n a n c e s in p - ~ e x p e r i m e n t s . We would like to thank P r o f e s s o r s S. Goldhaber, G. Goldhaber, and F. E. Low for helpful d i s c u s stons. After the conclusion of this w o r k we have l e a r n e d about a k i n e m a t i c a l M o n t e - C a r l o analysis of the A 1 peak done by the A a c h e n - B e r l i n - B i r m i n g h a m - Bonn- H a m b u r g - London- MOnc ~en Collaboration (preprint). 1. R.T.D~ck, Phys. Rev.Letters 13 (1964) 169. 2. G. Goldhaber, J . L . Brown, S. Goldhaber, J.A. Kodyk, B.C. Shen and G. H. Trill'.ng, Phys. Ray. Letters 12 (1964) 336. Aachen-Birmingham-Bonn-Hamburg-LondonM~nchenCollaboration, Ph~sics Letters 10 (1964) 226. 3. E. Ferrari and F. Selleri, Nuovo Cimento Suppl. 24 (1962) 453. 4. P.M.Ogden, Thesis, UCRL Report No. 11180 (1964). unpublished; J.A.Helland, T.J.Devlin, D.E .Hagge, M.J. Longo. B.J. Moyer andC.D. Wood, Phys. Rev. 134I~ (1964~ 1062. 5. G.Goldhaber, W.Chinowsky, S.Goldhaber, W.Lee and T.OWH:~lloran, Physics Letters 6 (1963) 62. 6. T.P.Wangler, A.R. Erwtn and W.D.Walker, Physics Letters 9 (1964) 71. 7. M. Abolins, R. L. Lander, W.A.W. Mehlhop, N.Xuong and P. M. Yager, Phys. Rev. Letters 11 (1963) 381. 8. C.Alff, D.Berley, D.Colley, N.Gelfand, U.Nauenberg, D.Miller, J.Schultz, J.Steinbcrger, T.H.Tan, tl. Brugger, P. Krarner and R. Plano, Phys. Rev. Letters 9 (1962) 3".~2W. D. Shephard, W.D. Walker, Phys. Rev. 126 (1962~ 278: Aachen-Btrmingham-Bonn-Hamburg-LondonM~nchen Coll at,oration, Nuovo C imento 31 (1964) 729. 283