- Email: [email protected]
Scaling of semi-inclusive distributions and the Feynman gas model
Nuclear Physics B46 (1972) 547-556. North-Holland Publishing Company
SCALING OF SEMI-INCLUSIVE DISTRIBUTIONS AND THE FEYNMAN GAS MODEL J. HUSKINS Physics Department, Imperial College, London SW7 Received 17 May 1972 Abstract: Scaling of semi-inclusive distributions proposed recently by Koba, Nielsen and Olesen is considered. We show that it does not follow just from scaling of the inclusive distributions and pionization. Specific examples are considered, including the Feynman gas model.
1. INTRODUCTION In a recent series of papers [ 1 - 3 ] Koba, Nielsen and Olesen (KNO) discuss the scaling properties of semi-inclusive distributions (inclusive distributions in which the prong number is not summed). They claim that if all inclusive distributions scale in the limit s ~ ~o, and approach a non-zero limit at x = 0 (x --- 2Pll/X/s), then [1, 2] (1)
Pn(s) =- an(S)/°T(S) ~ (n)
where n refers to the prong number, and
1 d3- o,,(s) - ~ h
d3p/ep
;x,p
T) .
(2)
It is shown in sect. 2 that the assumption of ordinary scaling, together with the nonzero limit of all distributions near x = 0 does not imply eqs. (1) and (2). The case where all inclusive distribution functions factorize is presented as a counter example to (1), though the result (2) holds. There is an experiment [4] where approximate scaling of the semi-inclusive distributions has been found, and consideration of simple non-factorizable models could be interesting. The Feynman gas model [5] is such an example, and is discussed in sect. 3.
J. Huskins, Semi-inclusive distributions
548
2. ASYMPTOTIC FORM OF THE DISTRIBUTION FUNCTIONS Consider the process a + b ~ c + (n - 1) charged + any number of neutrals,
(3)
where charge is not conserved. The semi-inclusive distribution function, gn(S; x, PT), for this process is defined by:
do n °n(S)gn(S;X, PT) - - - d3p/Ep
¢,o
_
1
(n - 1 ) ! k ~ 0 1 ! =
fd3p2 e2... n
X64(Pa+Pb-P-
d3pn d3q 1 G
el
d3q k --
e
k
~Pi
~
i=2
j=l
qj) lTk, n 12"
(4)
The set of equations [2]:
(n - 1) (n - 2 ) . . . (n - l ) ~ g n ( s , ' k ,
PT)
n=l+l d3p2.
l'O(s;x'PT'P2"''Pl)
E2
d3pl "" E l '
(5)
then follow simply from the definition o f f / , 0 : f/,0(p 1 . . . Pl) = 1/°T (cross section to produce charged tracks with momenta Pl - - • Pn + anything). KNO claim to invert this set of equations in an asymptotic approximation to obtain eq. (1) and (2). However, it is simple to perform the inversion at all energies, and then look at the asymptotic form. This task is most easily accomplished using the generating functional methods o f ref. [6]. As we are at no stage interested in observing neutral particles, the problem is equivalent mathematically to a system where a single type of particle is produced, here labelled charged * * All results apply to the more realistic case where positive, negative and neutral particles are produced if n c is read as the multiplicity of either the positive or negative tracks, the other charge type remaining unobserved, as are the neutrals.
J. Huskins, Semi-inclusive distributions
549
! The basic equation is [6]
/(be(P), q~o(q)) =
dane,n0 m
~ 1 n c,n o =0 nc!no!
f ~ P l • " 6Pn c 5ql " " fiqn 0
nc nO ~1 dP(pi) 6p i l-~ ¢(q/) 8q! i=1 /=l donc,nO ex = nc,n E 0 nc!no[ f 6 p l . . 6 P n c f q l . .
1
nc
5qno
no
× ~[ (1 + ¢(pi)) 8pi lq (1 + ~(q/)) 8q/, 1
(6)
1
where 5pi = d3pi/E i. In particular, we are only interested in
~(¢c(P); Co(q)
= o).
The set of eqs. (17) are then given by evaluating 6 fdP2 dp n ~ i~ I(¢c; 0), 8¢c(P) "'" ~¢c(P2) "''6dde(Pn)
(7)
in the limit ¢c(P) ~ O. The inverted equations are simply given by the limit ¢c(P) ~ - 1 , being k
Pr(s)
°r(s) _ 1 c~r+
=OT(S )
( - - l ) nc-r r!n = 1 (nc 2 r - - i ) !
ffnc;n°=O (s;xi, PiT) I-[ 8Pi, 1
(8) 1
g.(~; x, p 9 --Pn~) (~ - 1)!
m~ n (_ 1)m-n
nr
(m -~): f F ;0(~;~;, PiT) I-I @;. 2
(9)
J. Huskins, Semi-inclusive distributions
550
If one applies the result (14) of ref. [2]:
f f m ' O ( x, PT;P2 " • Pm) 6P2 -" " 6Pm =~m'0( x, PT; 0 . . . . 0) (ln s) m - 1 + 0 ((In s)n-2),
(10)
where
?m'O(x, pT; O, " " , O) = f fm'O(X, PT, P2 . . Pm) d 2 p 2 T ' ' d 2 p m T
'
the correction terms make the RHS meaningless due to the alternating nature of the series. However, even neglecting these corrections completely, the scaling properties clearly depend critically on the values of the f ( P l ; 0 . . . 0), so that one can not imply the scaling eqs. (1) and (2) without more input. It is trivial to evaluate Pn and gn from (8) and (9) in the completely factorizable case: m fm'O(xi, PiT: i= 1, m) = [--[ f l , O ( x i, PIT).
(11)
1
The multiplicity distribution is of course Poisson
Pn(s) :~-.t (nc(S))n e-
N / ~ ( ( n c ( S ) ! l n e n-
-~ ~
(12)
and the single-particle distribution function is
gn( s;x, PT) =___n_n (nc)~f l , 0 t ,x , PT)"
(13)
Eq. (1) is not satisfied, though eq. (2) is seen to hold.
3. FEYNMAN GAS MODEL In this model it is again possible to evaluate Pn and gn explicitly due to the relatively simple structure. The distribution functions are described by the single-particle
J. Huskins, Semi-inclusive distributions
55 1
distribution function C l(p; s) and the two-particle correlation function C2(Pl, P2;S) (ref. [6]). The model is not strictly compatible with energy-momentum conservation, which requires all correlation functions to be non zero in general. The final state particles may again be classified as charged (non-conserved) and neutral, the neutral particles not entering the mathematics, as we are not interested in observing them. As well as C 1 and C 2 there may be any number of other non-zero correlation functions, providing at least one of their arguments refers to a neutral particle. The correlations are taken to be of short range. Then,
C 1 =- fCl(s;p) d~pp ~alns,
C2 _~ rjc2(s;p, q) d3p Ep d3q l:q ~ h_ l n s ,
(14)
f o r S ---~co.
For convenience, C 2 will be taken negative (see appendix). The functions
C2 ( p , s ) =
a~2(s;p,q) d3q=
Eq
fC2(s; q, P) d3q
Eq'
and C l(p, s) are assumed to scale asymptotically:
Cl(P, s) ~ Ca(x,
PT), s ~ oo.
C2( p,
s) ~ C2(x,
(15)
PT),
The charge generating function of the model is the generating function for the Hermite polynomials [7], so that
n, Pn(s) = [(_~)n eClO+ ~C20~
_-
(,/_ ~c2)~
n e - C,+~C~ ~n(Z),
where z~~
1
0=_1
(C 1 - C2) ,
(16)
(17t
552
J. Huskins, Sembinclusive distributions
(n -- 1)! Pn(s) gn(P, s) = [~--~ n - 1 I~-~p) exp( f Cl(P) ¢(p) d3~p
6
+
f c2(P, q) ¢(P)
d3p d3
= ¢] (18)
1 n--1 LCC22~ N/_ IC 2 Un(z) = e- C1+~C2 (X/-2C2)
(C1 (P) C2(P)~ l(Z1 + C1 \ C 1 - C2 ] Hn_ •
(19)
The single-particle distribution function is then
rc2(x, pT) f,(x,pT)c2(x,pT)]~ gn(P's)=nE
-C-22
+~
C1
~
H'-l(Z)-1"
]
C1 Hn(Z) J
(2o)
The case when Cl(P) and C2(P) have the same shape is trivial, for then CI(P) C1
C2(P) C2 '
(21)
so that gn(P, S) = n
Cl(X, PT) n Cl(X, PT) ' C1 -
(22)
and (2) holds. In general, (21) will not be true, and the asymptotic scaling properties ofg n depend on the function R(n,(n))_CI
~ 2 C Hn-l(z) 2
for z ~ (In s)} ~ oo.
I-I,,(z)
~__ a H n - l ( z ) - v'~>
b
H,,(z)
'
(23)
J. Huskins. Semi-inclusive distributions
553
One is particularly interested in n ~ (constant) × In s and for three regions within this range there exist standard asymptotic forms for the Hn(z ) (ref. [7]) 1
(a) z = (2n + 1)~ cos ¢; 1
(b) z = (2n + 1)~ cosh ~; 1
1
1
1
(c) z = ( 2 n + l ) ~ - 2 -~ 3 -3 n -~t; real and 4~ and t fixed with z ~ ,~. 1
(a) z = (2n + 1~ cos ~n" Here, 1
1
1
e-~z~ Hn(Z ) = 2~ n + ~ (n!)~ (rm)-,i (sin qSn)-'I X [sin ((An +-~) (sin 2~ n - 2qSn) + ~Tr) + O(n-1)].
(24)
In evaluating R(n, (n)), care must be exercised in replacing (n - 1) and Cn- 1 by n and Cn: (-~n - ¼) (sin 2On_ 1 - 24~n_ 1) = (-~n +-~) (sin 2q~n - 24)n)
+ Cn + O(n-1),
(25)
giving
Hn-l(Z)
1 sin((½n +¼) (sin 2~bn - 2On) + q~n +37r) + O(n-l) (26)
Hn(z) - X / ~ 1
sin ((-~n +¼) (sin 2q)n - 2q~n) +3rr) + O(n -I) sin(~nO + ~ b + ~ T r ) + O ( n -1)
(27)
V~ffsin(-~ n 0 - ~ o + z l 3rr)+O(n-1 ) where
0 = sin 20 - 24), ~b= COS-1 (Z/N/-~), so that
Ra(n, (n))= r?a(n,¢)(V~n~ n) V_ -~. a ?--y
(28)
J. Huskins, Semi-inclusive distributions
554
In varying n at fixed q~(i.e. fLxed n/(n)) the numerator and denominator of r/a oscillate with the same frequency, but here is the extra phase q~in the numerator. Therefore, R a is a function ofn/(n) and n. 1
(b) z l
=
(2n
+
1)~ cosh q~n" Here,
2
3
1
I
e-~z Hn(z ) = 2 ½n-s (n!) ~ Qrn)-~ (sinh 4)n)q-
X exp ((~n + ~ (2q~n - sinh 2~bn)) (1 + O(n-1)).
(29)
Eq. (25) holds with sin ~b replaced by sinh ~ giving:
/4,,_ 1 (z)
1
Hd~)
V~
e -~n (l + O(n-l))
N / ~ (r~b
Rb(",
(30)
+ 0 (rt -1 )),
<.))-~IF-S~V_~ IV-h-nb ~h-<">.
(31)
(c) z = (2n + 1 ) { - 2-~ 3-~ n-k t n. Here, 1
3
1
1
_2
e-~x2 Hn(Z ) = 37 lr-r 2 ~n +~ (n!)~ ~ (A(tn) + O(n 7)),
(32)
where A(t) is Airy's function.
As t. = t . _ 1 + O ( n
_!
~),
* A preprint has recently been received from P. Olesen (NBI-HE-72-5) in which the Feynman model is discussed. His result (2.20) disagrees with ours, apparently through his treatment of the asymptotic expansions.
J. Huskins, Semi-inclusive distributions
555
The scaling ofgn(x, PT, s) therfore depends on the relative multiplicity, n/(n). For n/(n) = r(1 - b/a)2/( - 4b/a), one gets the semi-inclusive scaling of eq. (2) for r < 1, the scaling breaking * down for r > 1. In fact, for r > 1 the cross sections computed from C 1 and C 2 can become negative, so that the model becomes meaningless for such multiplicities. (This is not true for C 2 positive.) 4. CONCLUSIONS Although semi-inclusive scaling does not follow from inclusive scaling in the general case, it is seen to follow in the particular case of the gas model over some range of multiplicities. Allowing higher correlation functions then C 2 to be nonzero is unlikely tO improve this situation, unless these are carefully chosen; though the distributions of relatively low multiplicity may still scale if these correlations are not too large. The author wishes to thank Dr. I.G. Halliday for useful discussions and encouragemenL and the Science Research Council for financial support. APPENDIX For the gas model with C 2 positive, the asymptotic form ofgn(S, p) may be obtained from the integral representation for the Hermite polynomial
Hn(z ) = e z2 2 n+l 7r-~ f d t d-t2 t n cos ( 2 z t - ~mr). 0 For positive real z, z2/n finite, and n ~
Hn - 1(iz)
oo
- i
(A.1)
z +
Then, from eq. (-20), the semi inclusive distribution function becomes asymptotically:
Z
(C l (x, PT) - - ~ C2(x' PT))/3
1+
+ ~-~/T
where (A.2) and scaling is seen to hold for all finite relative multiplicities.
,
556
J. Huskins, Semi-inclusive distributions
REFERENCES [1] [2] [3] [4i [5] [6] [7]
Z. Koba, H.B. Nielsen and P. Olesen, NBI-HE-71-1. Z. Koba, H.B. Nielsen and P. Olesen, Phys. Letters 28B (1972) 25. P. Olesen, NBI-HE-72-1. P.V. Chliapiukov et al., CERN/D. Ph. II/PHYS, 72. K. Wilson, CLNS-131, Cornell, 1972, unpublished. L.S. Brown, Phys. Rev. D5 (1972) 748. Szego, Orthogonal polynomials, Am. Math. Soc. Collequium Publications, Vol, XXIlI (1959).
SCALING OF SEMI-INCLUSIVE DISTRIBUTIONS AND THE FEYNMAN GAS MODEL J. HUSKINS Physics Department, Imperial College, London SW7 Received 17 May 1972 Abstract: Scaling of semi-inclusive distributions proposed recently by Koba, Nielsen and Olesen is considered. We show that it does not follow just from scaling of the inclusive distributions and pionization. Specific examples are considered, including the Feynman gas model.
1. INTRODUCTION In a recent series of papers [ 1 - 3 ] Koba, Nielsen and Olesen (KNO) discuss the scaling properties of semi-inclusive distributions (inclusive distributions in which the prong number is not summed). They claim that if all inclusive distributions scale in the limit s ~ ~o, and approach a non-zero limit at x = 0 (x --- 2Pll/X/s), then [1, 2] (1)
Pn(s) =- an(S)/°T(S) ~ (n)
where n refers to the prong number, and
1 d3- o,,(s) - ~ h
d3p/ep
;x,p
T) .
(2)
It is shown in sect. 2 that the assumption of ordinary scaling, together with the nonzero limit of all distributions near x = 0 does not imply eqs. (1) and (2). The case where all inclusive distribution functions factorize is presented as a counter example to (1), though the result (2) holds. There is an experiment [4] where approximate scaling of the semi-inclusive distributions has been found, and consideration of simple non-factorizable models could be interesting. The Feynman gas model [5] is such an example, and is discussed in sect. 3.
J. Huskins, Semi-inclusive distributions
548
2. ASYMPTOTIC FORM OF THE DISTRIBUTION FUNCTIONS Consider the process a + b ~ c + (n - 1) charged + any number of neutrals,
(3)
where charge is not conserved. The semi-inclusive distribution function, gn(S; x, PT), for this process is defined by:
do n °n(S)gn(S;X, PT) - - - d3p/Ep
¢,o
_
1
(n - 1 ) ! k ~ 0 1 ! =
fd3p2 e2... n
X64(Pa+Pb-P-
d3pn d3q 1 G
el
d3q k --
e
k
~Pi
~
i=2
j=l
qj) lTk, n 12"
(4)
The set of equations [2]:
(n - 1) (n - 2 ) . . . (n - l ) ~ g n ( s , ' k ,
PT)
n=l+l d3p2.
l'O(s;x'PT'P2"''Pl)
E2
d3pl "" E l '
(5)
then follow simply from the definition o f f / , 0 : f/,0(p 1 . . . Pl) = 1/°T (cross section to produce charged tracks with momenta Pl - - • Pn + anything). KNO claim to invert this set of equations in an asymptotic approximation to obtain eq. (1) and (2). However, it is simple to perform the inversion at all energies, and then look at the asymptotic form. This task is most easily accomplished using the generating functional methods o f ref. [6]. As we are at no stage interested in observing neutral particles, the problem is equivalent mathematically to a system where a single type of particle is produced, here labelled charged * * All results apply to the more realistic case where positive, negative and neutral particles are produced if n c is read as the multiplicity of either the positive or negative tracks, the other charge type remaining unobserved, as are the neutrals.
J. Huskins, Semi-inclusive distributions
549
! The basic equation is [6]
/(be(P), q~o(q)) =
dane,n0 m
~ 1 n c,n o =0 nc!no!
f ~ P l • " 6Pn c 5ql " " fiqn 0
nc nO ~1 dP(pi) 6p i l-~ ¢(q/) 8q! i=1 /=l donc,nO ex = nc,n E 0 nc!no[ f 6 p l . . 6 P n c f q l . .
1
nc
5qno
no
× ~[ (1 + ¢(pi)) 8pi lq (1 + ~(q/)) 8q/, 1
(6)
1
where 5pi = d3pi/E i. In particular, we are only interested in
~(¢c(P); Co(q)
= o).
The set of eqs. (17) are then given by evaluating 6 fdP2 dp n ~ i~ I(¢c; 0), 8¢c(P) "'" ~¢c(P2) "''6dde(Pn)
(7)
in the limit ¢c(P) ~ O. The inverted equations are simply given by the limit ¢c(P) ~ - 1 , being k
Pr(s)
°r(s) _ 1 c~r+
=OT(S )
( - - l ) nc-r r!n = 1 (nc 2 r - - i ) !
ffnc;n°=O (s;xi, PiT) I-[ 8Pi, 1
(8) 1
g.(~; x, p 9 --Pn~) (~ - 1)!
m~ n (_ 1)m-n
nr
(m -~): f F ;0(~;~;, PiT) I-I @;. 2
(9)
J. Huskins, Semi-inclusive distributions
550
If one applies the result (14) of ref. [2]:
f f m ' O ( x, PT;P2 " • Pm) 6P2 -" " 6Pm =~m'0( x, PT; 0 . . . . 0) (ln s) m - 1 + 0 ((In s)n-2),
(10)
where
?m'O(x, pT; O, " " , O) = f fm'O(X, PT, P2 . . Pm) d 2 p 2 T ' ' d 2 p m T
'
the correction terms make the RHS meaningless due to the alternating nature of the series. However, even neglecting these corrections completely, the scaling properties clearly depend critically on the values of the f ( P l ; 0 . . . 0), so that one can not imply the scaling eqs. (1) and (2) without more input. It is trivial to evaluate Pn and gn from (8) and (9) in the completely factorizable case: m fm'O(xi, PiT: i= 1, m) = [--[ f l , O ( x i, PIT).
(11)
1
The multiplicity distribution is of course Poisson
Pn(s) :~-.t (nc(S))n e-
N / ~ ( ( n c ( S ) ! l n e n-
-~ ~
(12)
and the single-particle distribution function is
gn( s;x, PT) =___n_n (nc)~f l , 0 t ,x , PT)"
(13)
Eq. (1) is not satisfied, though eq. (2) is seen to hold.
3. FEYNMAN GAS MODEL In this model it is again possible to evaluate Pn and gn explicitly due to the relatively simple structure. The distribution functions are described by the single-particle
J. Huskins, Semi-inclusive distributions
55 1
distribution function C l(p; s) and the two-particle correlation function C2(Pl, P2;S) (ref. [6]). The model is not strictly compatible with energy-momentum conservation, which requires all correlation functions to be non zero in general. The final state particles may again be classified as charged (non-conserved) and neutral, the neutral particles not entering the mathematics, as we are not interested in observing them. As well as C 1 and C 2 there may be any number of other non-zero correlation functions, providing at least one of their arguments refers to a neutral particle. The correlations are taken to be of short range. Then,
C 1 =- fCl(s;p) d~pp ~alns,
C2 _~ rjc2(s;p, q) d3p Ep d3q l:q ~ h_ l n s ,
(14)
f o r S ---~co.
For convenience, C 2 will be taken negative (see appendix). The functions
C2 ( p , s ) =
a~2(s;p,q) d3q=
Eq
fC2(s; q, P) d3q
Eq'
and C l(p, s) are assumed to scale asymptotically:
Cl(P, s) ~ Ca(x,
PT), s ~ oo.
C2( p,
s) ~ C2(x,
(15)
PT),
The charge generating function of the model is the generating function for the Hermite polynomials [7], so that
n, Pn(s) = [(_~)n eClO+ ~C20~
_-
(,/_ ~c2)~
n e - C,+~C~ ~n(Z),
where z~~
1
0=_1
(C 1 - C2) ,
(16)
(17t
552
J. Huskins, Sembinclusive distributions
(n -- 1)! Pn(s) gn(P, s) = [~--~ n - 1 I~-~p) exp( f Cl(P) ¢(p) d3~p
6
+
f c2(P, q) ¢(P)
d3p d3
= ¢] (18)
1 n--1 LCC22~ N/_ IC 2 Un(z) = e- C1+~C2 (X/-2C2)
(C1 (P) C2(P)~ l(Z1 + C1 \ C 1 - C2 ] Hn_ •
(19)
The single-particle distribution function is then
rc2(x, pT) f,(x,pT)c2(x,pT)]~ gn(P's)=nE
-C-22
+~
C1
~
H'-l(Z)-1"
]
C1 Hn(Z) J
(2o)
The case when Cl(P) and C2(P) have the same shape is trivial, for then CI(P) C1
C2(P) C2 '
(21)
so that gn(P, S) = n
Cl(X, PT) n Cl(X, PT) ' C1 -
(22)
and (2) holds. In general, (21) will not be true, and the asymptotic scaling properties ofg n depend on the function R(n,(n))_CI
~ 2 C Hn-l(z) 2
for z ~ (In s)} ~ oo.
I-I,,(z)
~__ a H n - l ( z ) - v'~>
b
H,,(z)
'
(23)
J. Huskins. Semi-inclusive distributions
553
One is particularly interested in n ~ (constant) × In s and for three regions within this range there exist standard asymptotic forms for the Hn(z ) (ref. [7]) 1
(a) z = (2n + 1)~ cos ¢; 1
(b) z = (2n + 1)~ cosh ~; 1
1
1
1
(c) z = ( 2 n + l ) ~ - 2 -~ 3 -3 n -~t; real and 4~ and t fixed with z ~ ,~. 1
(a) z = (2n + 1~ cos ~n" Here, 1
1
1
e-~z~ Hn(Z ) = 2~ n + ~ (n!)~ (rm)-,i (sin qSn)-'I X [sin ((An +-~) (sin 2~ n - 2qSn) + ~Tr) + O(n-1)].
(24)
In evaluating R(n, (n)), care must be exercised in replacing (n - 1) and Cn- 1 by n and Cn: (-~n - ¼) (sin 2On_ 1 - 24~n_ 1) = (-~n +-~) (sin 2q~n - 24)n)
+ Cn + O(n-1),
(25)
giving
Hn-l(Z)
1 sin((½n +¼) (sin 2~bn - 2On) + q~n +37r) + O(n-l) (26)
Hn(z) - X / ~ 1
sin ((-~n +¼) (sin 2q)n - 2q~n) +3rr) + O(n -I) sin(~nO + ~ b + ~ T r ) + O ( n -1)
(27)
V~ffsin(-~ n 0 - ~ o + z l 3rr)+O(n-1 ) where
0 = sin 20 - 24), ~b= COS-1 (Z/N/-~), so that
Ra(n, (n))= r?a(n,¢)(V~n~ n) V_ -~. a ?--y
(28)
J. Huskins, Semi-inclusive distributions
554
In varying n at fixed q~(i.e. fLxed n/(n)) the numerator and denominator of r/a oscillate with the same frequency, but here is the extra phase q~in the numerator. Therefore, R a is a function ofn/(n) and n. 1
(b) z l
=
(2n
+
1)~ cosh q~n" Here,
2
3
1
I
e-~z Hn(z ) = 2 ½n-s (n!) ~ Qrn)-~ (sinh 4)n)q-
X exp ((~n + ~ (2q~n - sinh 2~bn)) (1 + O(n-1)).
(29)
Eq. (25) holds with sin ~b replaced by sinh ~ giving:
/4,,_ 1 (z)
1
Hd~)
V~
e -~n (l + O(n-l))
N / ~ (r~b
Rb(",
(30)
+ 0 (rt -1 )),
<.))-~IF-S~V_~ I
(31)
(c) z = (2n + 1 ) { - 2-~ 3-~ n-k t n. Here, 1
3
1
1
_2
e-~x2 Hn(Z ) = 37 lr-r 2 ~n +~ (n!)~ ~ (A(tn) + O(n 7)),
(32)
where A(t) is Airy's function.
As t. = t . _ 1 + O ( n
_!
~),
* A preprint has recently been received from P. Olesen (NBI-HE-72-5) in which the Feynman model is discussed. His result (2.20) disagrees with ours, apparently through his treatment of the asymptotic expansions.
J. Huskins, Semi-inclusive distributions
555
The scaling ofgn(x, PT, s) therfore depends on the relative multiplicity, n/(n). For n/(n) = r(1 - b/a)2/( - 4b/a), one gets the semi-inclusive scaling of eq. (2) for r < 1, the scaling breaking * down for r > 1. In fact, for r > 1 the cross sections computed from C 1 and C 2 can become negative, so that the model becomes meaningless for such multiplicities. (This is not true for C 2 positive.) 4. CONCLUSIONS Although semi-inclusive scaling does not follow from inclusive scaling in the general case, it is seen to follow in the particular case of the gas model over some range of multiplicities. Allowing higher correlation functions then C 2 to be nonzero is unlikely tO improve this situation, unless these are carefully chosen; though the distributions of relatively low multiplicity may still scale if these correlations are not too large. The author wishes to thank Dr. I.G. Halliday for useful discussions and encouragemenL and the Science Research Council for financial support. APPENDIX For the gas model with C 2 positive, the asymptotic form ofgn(S, p) may be obtained from the integral representation for the Hermite polynomial
Hn(z ) = e z2 2 n+l 7r-~ f d t d-t2 t n cos ( 2 z t - ~mr). 0 For positive real z, z2/n finite, and n ~
Hn - 1(iz)
oo
- i
(A.1)
z +
Then, from eq. (-20), the semi inclusive distribution function becomes asymptotically:
Z
(C l (x, PT) - - ~ C2(x' PT))/3
1+
+ ~-~/T
where (A.2) and scaling is seen to hold for all finite relative multiplicities.
,
556
J. Huskins, Semi-inclusive distributions
REFERENCES [1] [2] [3] [4i [5] [6] [7]
Z. Koba, H.B. Nielsen and P. Olesen, NBI-HE-71-1. Z. Koba, H.B. Nielsen and P. Olesen, Phys. Letters 28B (1972) 25. P. Olesen, NBI-HE-72-1. P.V. Chliapiukov et al., CERN/D. Ph. II/PHYS, 72. K. Wilson, CLNS-131, Cornell, 1972, unpublished. L.S. Brown, Phys. Rev. D5 (1972) 748. Szego, Orthogonal polynomials, Am. Math. Soc. Collequium Publications, Vol, XXIlI (1959).