Phase space effects on single particle spectra

Volume 38B, number 4

PHASE

PHYSICS L E T T E R S

SPACE

EFFECTS

ON S I N G L E

21February 1972

PARTICLE

SPECTRA*

M. -S. CHEN and F. E. PAIGE

Brookhaven National Laboratory, Upton, New York 11973, USA Received 15 November 1971

We investigate whether the single particle spectra should behave hke distributmns m one- or threedlmensmnal phase space. From a statmtical model and the observed momentum dmtmbutmn, we demonstrate that the phase space is essentmlly one-dimensmnal for Plab >~ 10-20 GeV/c.

In a r e c e n t l e t t e r B e r g e r and Krzywicki [1] suggest that i n c l u s i v e r e a c t i o n s should exhibit two c h a r a c t e r i s t i c energy r e g i m e s . They state that m the lower e n e r g y r e g i m e , Plab ~< 100 GeV/c, the a v e r a g e s e c o n d a r y - p a r t i c l e m o m e n tum is sufficmntly s m a l l that the t r a n s v e r s e mom e n t u m cutoff has little effect. Thus the t h r e e d i m e n s m n a l s t a t i s t i c a l model should be approp r i a t e ; this gives an a v e r a g e multiplicity m c r e a s i n g like s 1/3. Only in the higher energy r e g i m e should the t r a n s v e r s e m o m e n t u m cutoff play a m a j o r role, making the o n e - d l m e n s m n a l s t a t m t i c a l model applicable. This r e s u l t s in an a v e r a g e m u l t i p l i c i t y i n c r e a s i n g like log s. The t r a n s i t i o n r e g i o n between these two r e g i m e s occupies a span of s e v e r a l hundred GeV. In what follows an approximate, a n a l y t i c a l l y solvable v e r s i o n of the s t a t i s t i c a l model with a t r a n s v e r s e m o m e n t u m cutoff is considered. It is shown that the a v e r a g e multiplicity in this model has an expansion of the form /~(s) = c logs+d+c' logs/s + d'/s +...

(1)

and that the f i r s t two t e r m s provide a good app r o x i m a t i o n for Plab > 10-20 GeV/c. Some exp e r i m e n t a l single p a r t i c l e s p e c t r a a r e also exa m i n e d and shown to support the o n e - d i m e n s i o n a l a p p r o x i m a t i o n at s i m i l a r e n e r g i e s . In the s t a t i s t i c a l model the p r o b a b i l i t y for producing N p a r t i c l e s of m a s s m with total mom e n t u m P is taken to be p r o p o r t i o n a l to [2] ~tN

y., aN(P) ~tN

N

3 d p~

=NT• f i~=l ~ l - 2 f () P/ i ± ) 6 " ( = 2(pz+m

a

..N

(2)

where the functions f(Pi±) provide the t r a n s v e r s e m o m e n t u m cutoff. While the a v e r a g e m u l t i p h c i t y can, in p r i n c i p l e , be obtained d i r e c t l y from eq. (2), the i n t e g r a l s cannot be done a n a l y t i c a l l y for N >/ 3. To obtain an approximate solution, it is convenient to introduce an a u x i l i a r y functmn [3] oo ~ N

(2(~,a) =N~2~. f d4p exp ( - a

(3) N=2 ~.. [q~(~)]N ~ exp (hq~(o~)) where ~ is a t i m e l i k e f o u r - v e c t o r and

¢(a) : f

d3p

2(p2+m2)l/2

f(p±) exp (-a. p).

(4)

The function Q (X, ~), the analogue of the g r a n d p a r t i t i o n function in s t a t i s t i c a l m e c h a n i c s , des c r i b e s an e n s e m b l e of s y s t e m s with v a r y i n g N and P. The a v e r a g e s of these quantities over the e n s e m b l e a r e given by [3] N = X~-~logQ(X, ot),

P

=---

logQ(X,~).(5)

on Bye±andL°rentz~.,2~0invariance_ ~,, 2)1/29(~)in theeanframedepend~,,°nly= 0, the p, i n t e g r a t i o n in eq. (4) can be done, giving [2] (6) oO

~(a) = an

f dpidp±f(p ±) Io(a±p±) Ko(~(p±2+m2)l/2). 0

In the c e n t e r of m a s s f r a m e , /~= (W,0), eqs. (5) and (6) yield

: 2nk f d p ~ p z f ( p . ) go(fl(p±2 +m2)1/2)

p-~-j i=l Pi ),

* Work performed under the auspmes of U.S. Atomm Energy Commissmn.

" P) fiN(P)

0 _

~o

(7) 2

2 1/2

W= 2n~ f dp±p~(p. +m ) 0

f(p±)gl(~(p

2

2

1/2

+m )

249

)

Volume 38B, number 4

PHYSICS

and at_ = O. In t h e s e e q u a h o n s l a r g e W- c o r r e s p o n d s to s m a l l B, Eq. (7) actually glves the average multiphctty of an ensemble of systems with various values of W. One can easily verify that in the limit W~ ~,

1/2c

( ~ 2 _ W2)/W 2 ~

LETTERS

Table 1 Compar,son of the average mulhplicltms obtained from the exact formula and the asymptohc expressmn as a functmn of energy

(8)

w h e r e c is the c o e f h c i e n t of l o g W 2 ~n the a v e r a g e mulhplicity, ~V

c log ~ 2

+ ...

(9)

H o w e v e r , the v a l u e of c a g r e e s with the c o e f f i c i e n t of log s in the a v e r a g e m u l t i p l i c i t y for the s t a t l s h c a l m o d e l with a t r a n s v e r s e m o m e n t u m cutoff [2]; eq. (7) a l s o r e p r o d u c e s the c o r r e c t a s y m p t o t i c m u l h p h c t t y w h e n the cutoff is r e m o v e d [3 ]. It is t h e r e f o r e r e a s o n a b l e to i d e n t i f y W with the a c t u a l e n e r g y W and to u s e (7) as a m o d e l for the d e p e n d e n c e of N on IV. If f(p~) d e c r e a s e s f a s t e r than any p o w e r of p±, then the a s y m p t o t i c e x p a n s m n of f f tn t e r m s of W can be o b t a i n e d f r o m eq. (7) by using the s e r i e s K 0 ( f i ( p ~ + r n 2 ) l / 2 ) and Kl(fi(p±2+m2)l/2), i n t e g r a t i n g t e r m by t e r m , and e h m i n a h n g 3. The r e s u l t is

21 February 1972

W

Nexac t

1.20 2.06 3.01 4.34 5.37 6.98 9.92

2.17 2.98 3.63 4.30 4.70 5.20 5.88

N(s) = C logS~ + d + . ~ s

m2

Nasymp 1.63 2.71 3.47 4.21 4.63 5.16 5.86

ai(0)- I

~"

s

[ctl°gm.~+ diI, (12)

w h e r e a t a r e the n o n - l e a d i n g R e g g e t r a j e c t o r i e s . It s e e m s r e a s o n a b l e that d a u g h t e r t r a j e c t o r i e s of the P o m e r a n c h u k o n should c o n t m b u t e , and t h e s e g w e a f o r m for -~(s) v e r y s i m i l a r to that found in the s t a t i s t m a l m o d e l . (The only d m c r e p a n c y is the a p p e a r a n c e of h i g h e r p o w e r s of log s, s u c h a s the log 2 W/W 4 t e r m s in eq. (10).) In a d dxtmn the M u e l l e r f o r m a l i s m will, ~n g e n e r a l , include contmbuhons from meson trajectomes f f ~ a01og (2W/aom) + with a i ( 0 ) ~ 0.5, and t h e s e should d o m i n a t e the c o n t r i b u t m n s of the P o m e r a n c h u k o n d a u g h t e r s . + bo +Ta,~a~ u (m2/W 2) log(2W/aorn ) + W i t h i n the s t a t i s h c a l m o d e l , the f i r s t two (lO) t e r m s in eq. (1) a r e e x p e c t e d to g i v e a good a p 2 3 2 2 ~ W 2 ) _ 5a3oa211og2(2W/aom)+ p r o x l m a h o n to the a v e r a g e m u l t i p h c l t y , s i n c e + (4, aoal+~ao bl)(m T the next t e r m ts of o r d e r (~2+m2)W-21og(W/rn). T h i s can be m a d e m o r e p r e c i s e by u s i n g for the 1 2 5 4 ~ 3 , ,, 4/W 4) log(2W/aOm) + t r a n s v e r s e m o m e n t u m cutoff f ( p ± ) the o b s e r v e d + (~ a30 a l + 6 ~ a 0 a2-¥aoalol~( m s i n g l e - p a r h c l e s p e c t r u m [5]* , 1

4

1 3

,

~ 3,2

+ t l - ~ a 0 a2+~a 0 alOl-~aoO

5

4,

,,

4-

4

l+~aoO2)~rn /W ) + ...

p± 2+m2

)k f(p±), 0 (p ±2+m2)l/2 ~dp±p± log

m

~[Vt(k)+gJ(k+l)]a k-

27T)~f dp±p± ( P ± 2 + r n 2 ) 0

\

= 0.5772...,

m2

(11) (p±2+m2)

k log

rn

~P(k+l) = 1 + ~ + . . .

f(p±), k 1

+~-y.

By c o m p a r i s o n M u e l l e r ' s o p t i c a l t h e o r e m a p p r o a c h [4] g i v e s for the a v e r a g e m u l h p h c i t y

250

a = 10 GeV - 2

-~asymp. = 2 l o g WGe V = 1.27. f(P£) ,

0 bk :

= exp(-aP_L2),

(13)

The r e m a i n i n g p a r a m e t e r , ~, is a d j u s t e d to m a k e N(s) ~ 1.0 log s + . . . , as is e x p e m m e n t a l l y o b s e r v e d [6,7]. F r o m eqs. (10) and (11) the a s y m p t o h c e x p r e s s i o n for the m u l h p h c i t y is then

where

b0 : - Y a o - 2 ~ t

f(p±)

>0,

(14)

In t a b l e 1 thts e x p r e s s i o n is c o m p a r e d with the e x a c t r e s u l t o b t a i n e d by c o m p u h n g the i n t e g r a l s in eq. (7) n u m e r i c a l l y for a p p r o p r i a t e v a l u e s of ~. It m a c c u r a t e w i t h i n f i v e p e r c e n t for W > 3 GeV and wzthir, t e n p e r c e n t f o r W > 2 GeV. B e r g e r and K r z y w i c k i [1] p o i n t out that m p r o t o n - p r o t o n r e a c t i o n s the final s t a t e p r o t o n s a r e l e a d i n g p a r t i c l e s and h a v e an a v e r a g e e l a s t i c i t y 2p ~ 0.5. H e n c e the e f f e c t i v e v a l u e of W ~s * We only need an approximate form for the gross feature of the observed t r a n s v e r s e momentum distmbutmn (cf. ref. [5]).

Volume 38B, n u m b e r 4

PHYSICS

LETTERS

21 F e b r u a r y 1972 p+p --.-)"rr-+ANYTHING

p+p ~'tr++ ANYTHING i

i

i

05

0.51-

04

04

'

OE

,

,.~

..-~- Z ~ / t

04

-

04.

o

o o.3LI --- -

,~ O~

L9

.k"" 0.2

015

-

0.~

.4"

/

} ~

4. PRONGS 6 PRONGS 8 PRONGS

i...~_ ~

>03

CD

/ "/{ 02~ ' 0"15 |~_

02[.-I ~Z~'.-J ~4.6PRONGS

~4. PRONGS ,~6 PRONGS ~8 PRONGS

0

15

, Ro,.,os

L

i

,5

i

15

°sPR°'°s i 'Jo ~5 ~o

02F

l 0.15

: 4 PRONGS ..

P.ONOS

°,,P,,o,,,os ,5

2'0 ~ ~'o

BEAM MOMENTUM(GeV/c)

BEAM MOMENTUM (GeV/c)

Fig. io Log-logplots of the average longltudmal and transverse momentum versus incident beam energy for various topologles in pp --~Tr± + anything obta,ned from ref. [8]. less than s 1/2. One expects to see the maximum deviation from the one-dimensional approxxmation for events with higher than average multiplictty, for which the proton elasticity should be smaller than average. If the one-dimensional approximation is valid at some value of W, it should therefore be equally valid at a value of s no greater than s = W2(1 _~p)-2 ~ 4W2 "

(15)

Thus W = 3 GeV, for whlch the correchons to the asymptotic expression are less than five percent, corresponds to some Plab less than 18 GeV/c. (The correction at this energy due to Mueller graphs [2] involving meson exchanges or to other dynamical effects may, of course, be large.) The numerical difference between s I/3 and log s increase of multiplicity is actually quite small over a reasonable energy range. Thus we further study the difference between the onedimensional and three-dimenslonal regimes by comparing the distribution of the momentum of a secondary parhcle between longitudinal and transverse components. In the one-dimensional regime the average transverse momentum, p±, for an exclusive process should stay constant, and the average longitudinal momentum, p,,, should increase like s 1/2. In the three-dlmenslonal regime, on the other hand, the distribution should be isotroptc with ~ = f2~,,, reflechng the__two transverse degrees of freedom, and both p, and p-~ should increase like s 1/2. Similar behavior is also expected for inclusive distributions **. ** In the scahng limit, it can be shown that p± for the melus~ve dtstrlbution has a weak energy dependence of the form (p±> =¢o× (1 - eonstant/log(Plab/mp)) (cf. ref. [6]).

The transverse and longitudinal pton momentum distributions for four, six, and eight prong topologies have been measured tn proton-proton colhstons at 13, 18, 21, 24 and 28 GeV/c beam momenta by Smith et al. [8]. The p± and p,, obtained from their data are plotted vers us Plab on a log-log scale in fig. I. Since the topologlcal dtstribuhons are not the exclusive distributions wlth fixed multiplicihes, both p,, and p± are expected to increase slower than sl/2 even tn the three-dtmenslonal regime because there are more partmles produced at higher energies. But the observed increase of p,_ ts much greater than that of pj_. Furthermore, p~ stays nearly constant and is less than v~ p, even for the smallest of energy. The data thus indicate that the one-dtmenslonal descriptlon Is qualitatively correct in this energy range. To conclude, we have demonstrated from both the approximate statistical model and the observed momentum dlstributtons that the one-dimensional regime is qualitatively correct for P l a b ~ 1 0 - 20 G e V / c . H o w e v e r , t h e d y n a m i c a l corrections may be important so that the single particle spectra can deviate substantially from a stahstical one-dimensional phase space distribution. T h e a u t h o r s w o u l d t h a n k D r . R. F. P e i e r l s his crihcal comments.

for

References [1] E. L. B e r g e r and A. Krzywmkl, Phys. Letters 36B (1971) 380. [2] A.Krzywlckl, Nuovo Clmento 32 (1964) 1067. [3] H. Satz, Nuovo C~mento 37 (1965) 1407. [4] A. H. M u e l l e r , Phys. Rev. D2 (1970) 2936.

251

Volume 38B, number 4

PHYSICS

[5] E. W. Anderson et al., Phys. Rev. Letters 21 (1968) 830, J. L. Day et al., Phys. Rev. L e t t e r s 23 (1969) 1055: J. V. Allaby et al., High energy particle s p e c t r a from proton interaction at 19.2 GeV/c, CERN 70-12 (1970).

252

LETTERS

21 F e b r u a r y 1972

[6] N. F. Bah et al., Phys. Rev. L e t t e r s 25 (1970) 557. [7] L. W. Jones et al., Phys. Rev. L e t t e r s 25 (1970) 1679; D. E. Lyon et al., Phys. Rev. L e t t e r s 26 (1971) 728. [8] D. B. Smith et al., Phys. Rev. L e t t e r s 23 (1969) 1064.