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Scaling of semi-inclusive cross sections with respect to longitudinal and transverse momenta
Nuclear Physics B97 (1975) 219-226. © North-Holland Publishing Company
SCALING OF SEMI-INCLUSIVE CROSS SECTIONS WITH RESPECT TO LONGITUDINAL AND TRANSVERSE MOMENTA N.S. ANGELOV, N.N. MELNIKOVA, A.N. SOLOMIN * and L.M. SCHEGLOVA * Joint Institute for Nuclear Research, Dubna N.D. AKHABABYAN, L.D. GREKOVA, V.N. PENEV and A.I. SHKLOVSKAYA Institute for Nuclear Research and Nuclear Energy, Sofia Received 10 April 1975 (Revised 7 July 1975) The semi-inclusiveone-particle transverse and longitudinal momenta spectra of n± mesons have been studied in the centre-of-mass system for 7r-p interactions at 40 GeV/c. These data are compared with similar data from pp interactions at 300 GeV/c and lower energies. Distributions in the variables pT/(PT) and pL/
The analysis of the centre-of-mass inclusive single-particle distributions in the transverse (PT) and the longitudinal (PL) m o m e n t u m variables [1 ] reveals the following scaling property:
1
d2o
1
o dPT dPL = (pT) (PL) f ( P T / ( P T ) ' PL/(PL)) '
(1)
where f is a function, which is independent of multiplicity, incident energy and initial state. This implies that the dynamics are manifest only in the average values (PT) and (PL). In this paper we test this scaling behaviour hypothesis for 7r-p interactions at 40 GeV/c. The data were obtained from the Dubna 2m propane bubble chamber, exposed to a 40 GeV/c rr- beam from the Serpukhov accelerator. We have examined 6275 events of zr-p interactions at 40 GeV/c. These events were selected according with the criteria described in [2]. The one-particle spectra of charged secondary particles for the semi-inclusive process
* Institute for Nuclear Physics Research of Moscow State University.
N.S. Angelov et al. /Semi-inclusive cross sections
220
7r- + p ~ 7r-+ + ( n - - 1) charged particles + neutral particles
(2)
were m e a s u r e d as f u n c t i o n s o f the variables pT/(PT) and pL/(PL). P r o t o n s w e r e i d e n t i f i e d b y i o n i z a t i o n up t o Plab ~ 0.8 G e V / c . All t h e negative particles w e r e t a k e n as pions. In such a p r o c e d u r e the positive particles w e r e c o n t a m i n a t e d b y 5% K + a n d ~ + , and b y ~ 1 6 % n o n - i d e n t i f i e d p r o t o n s [ 3 ] . Table 1 The mean values of longitudinal and transverse momenta for n +- mesons for different topologies in 7r-p interactions at 40 GeV/c nch n
+
2
4
6
8
back forward
0.980 -+ 0.038 0.949 -+ 0.062
0.692 -+ 0.017 0.735 -+ 0.019
0.550 -+ 0.012 0.602 -+ 0.013
0.479 -+ 0.012 0.454 -+ 0.012
back forward
0.493 -+ 0.045 2.006 -+ 0.047
0.555 -+ 0.018 1.121 -+ 0.021
0.503 -+ 0.013 0.746 -+ 0.014
0.410 +- 0.011 0.535 -+ 0.013
7r+ 7r-
0.440 -+ 0.010 0.399 -+ 0.010
0.401 -+ 0.005 0.382 -+ 0.005
0.389 -+ 0.004 0.375 -+ 0.004
0.373 +- 0.004 0.362 -+ 0.005
10 + 12
~> 14
Inclusive
~p~ It
(P±) nch + rr
back forward
0.448 -+ 0.014 0.381 -+ 0.012
0.434 -+ 0.035 0.283 -+ 0.026
0.569 -+ 0.007 0.559 -+ 0.007
~r-
back forward
0.395 +- 0.013 0.423 -+ 0.013
0.684 +- 0.236 0.335 -+ 0.30
0.472 + 0.011 0.838 -+ 0.009
0.363 -+ 0.006 0.341 + 0.005
0.324 +- 0.015 0.301 -+ 0.012
0.384 -+ 0.002 0.367 -+ 0.002
+
(P±)
~r 7r-
Table 1 s h o w s the average values o f transverse and l o n g i t u d i n a l m o m e n t a in c.m.s. for e a c h m u l t i p l i c i t y o f charged s e c o n d a r y particles in n - p i n t e r a c t i o n s . These values have b e e n o b t a i n e d b y averaging the transverse and longitudinal m o m e n t a w i t h i n t h e k i n e m a t i c ranges d e t e r m i n e d for e a c h m u l t i p l i c i t y , taking into a c c o u n t t h e measurem e n t errors. Fig. 1 (a, b) illustrates t h e ((pT)/On) don/dPT shapes for d i f f e r e n t c h a n n e l s o f r e a c t i o n (2) for n - and n + m e s o n s , respectively. One can observe a universal behaviour o f t h e s e d i s t r i b u t i o n s , w h i c h are i n d e p e n d e n t o f nch. The d a t a w e r e fitted in c o m m o n for d i f f e r e n t nch t o t h e f u n c t i o n [ PT ~ c
(3)
A
(a)
.':
"''i:.
a5
Pl
65
o .
.~-
(b)
PJ. ~p.~> ~,+ Fig. 1. Plot o f ((pT>/On) dan~alp T versus p T / ( P T ) for t h e r e a c t i o n s 7r-p ~ 7r- (a) and 7r-p ~ 7r+ (b) for d i f f e r e n t n c h at 40 GeV/c. T h e solid curve p r e s e n t s t h e fit o f 7r-p d a t a at 40 G e V / e . The d a s h e d curve is t h e fit results o f t h e e x p e r i m e n t a l p p d a t a [1 ]. Table 2 T h e p a r a m e t e r s o f t h e universal f u n c t i o n s (3) and (4), d e s c r i b i n g in c o m m o n the 7r+ m e s o n distributions: (PT) b~ *
*
do n dp T '
(PL) d a n
on
dp L
k
p
q
x2/(N-M)
1.14 + - 0.03
1.27 + - 0.04
-0.05 + - 0.01
1.14
7¥
1.18 -+ 0 . 0 4
1.34 -+ 0.05
- 0 . 0 7 -+ 0.01
1.31
nB
1.20 + 0.04
1.36 + 0 . 0 4
- 0 . 0 7 -+ 0.01
1.37
7rF
1.17 + 0 . 0 4
1.31 +- 0 . 0 4
- 0 . 0 6 + 0.01
1.89
7rBa)
0.91 + 0.15
0.83 + 0 . 0 4
0.03 +-0.01
0.96
p.L/(p±)
a
c
b
x2/(N-M)
~+ ~r-
5.41 -+ 0.29 5.21 +- 0.27
1.14 _+ 0 . 0 4 1.14 -+ 0 . 0 4
2.28 -+ 0 . 0 4 2.25 -+ 0 . 0 4
1.06 1.25
~r-
6.23 -+ 0.52
1.37 +- 0.03
2.37 -+ 0.04
1.34
PlI/(PII ) 7r + B +
a) p p d a t a , see ref. [ 1 ] .
~ b ¢.0 09" 08" 0.7@~0.5" at"
t
a2 0A
s
"~,, •
~5
¢.5
i~k
.
2,5
35
~5 < p,*>
~05
" "~'
t
O.t °. ,
O.5
~5
#,5
,:15
45 II.
Fig. 2. Plot of (for the reaction ~r-p ~ ~r+ in the regions PL > 0 (Try) and PL < 0 (TrB) for different nch at 40 GeV/c. The solid curve presents the fit of n - p data at 40 GeV/c. The dashed curve is taken from ref. [1].
O8
g7
O5" 0~03-
0.~
t
0~
~
~
.15
i
I
o,,"
~5
f.t
Qg
a7"
"~
0.5
~x
0/,/
•
~,\
03
OZ Ill i
;
Fig. 3. Plot of ((pL)/On) don/dPL versus pL/(PL) for the reaction 7r-p ~ I t - in the regions PL > 0 (~F-) and PL < 0 (~B-) for different rich at 40 GeV/c. The dashed curve is taken from ref. [11.
N.S. Angelov et al. /Semi-inclusive cross sections
224
Table 3 The values of mean square deviations of the universal functions (3) and (4) with fixed parameters (see table 2) from the 40 GeV/c data at different nch nch
P±
2
4
6
8
10,12
/>14
Inclusive
7r-t-
back forward
1.19 1.20
1.10 1.09
1.15 1.79
1.41 2.47
1.01 1.15
1.63 0.97
2.54 2.07
7r-
back forward
1.01 7.75
0.76 8.66
1.52 2.20
1.33 1.90
0.60 0.92
1.31 0.61
1.45 20.76
1.16 1.51
1.29 0.83
2.07 1.70
1.38 1.86
1.30 1.60
0.86 0.53
3.57 2.76
~r+ n
and the universal parameters a, c and b were obtained (see table 2). For comparison the corresponding universal parameters for pp interactions [1 ] are also given in table 2. The dashed curve in fig. 1 corresponds to the data o f ref. [1 ]. The solid curve is obtained by fitting the data from 7r-p interactions at 40 GeV/c. As is seen from fig. 1, b o t h curves fit our data well within the experimental errors. In order to check to what extent our data is described b y the universal functions with the parameters obtained b y fitting the semi-inclusive data in common, we calculated the mean square deviations o f these functions from experimental distributions at different rich. The result is presented in table 3. It is seen that ((pT)/On) dan/dPw distributions are well described b y the universal function (3). To analyse the scaling properties o f the longitudinal momentum distributions, we have plotted the spectra ((pL)/On) don/dPL for the regions PL > 0 and PL < 0 separately. These spectra are given in figs. 2(a, b) and 3(a, b) for 7r+ and n - mesons respectively. The universal functions obtained by fitting the ~r- p data at 40 GeV/c (solid curve) and the universal function from paper [1 ] (dashed curve) are also shown in figs. 2 and 3. The experimental distributions for different rich were fitted in common to the following function
tpL = k exp
p
P((~L)) [ PL ]2] - qt(-~L)j J.
(4)
The parameters k, p and q are given in table 2. One can see from figs. 2 and 3, as well as from table 3, that the experimental data on 71.+ (sOL <~ 0) and 7r- (PL < 0) are well described b y the function (4) within the experimental errors. Some deviations from the universal behaviour come from the events with low multiplicity o f charged secondary particles or due to large measurement errors. The distributions for 7r- (PL > 0) (see fig. 3) are dominated b y a contribution from leading 7r- mesons for which one should not expect the scaling behaviour o f
P±
forward
zr-
7r +
back
forward
back
7r-
~r
+
~+
nch
Table 4 The results of the fit of
a c b x2
4.02 -+ 0.55 0 . 9 4 _+ 0.09 2.07 -+ 0.11 1.33
6,57 -+ 1.18 1.35 _+ 0.13 2.45 _+ 0.15 1.02
0.60 ± 0.06
k
a c b x2
1.38_+0.21 1.84 _+ 0.25 - 0 . 1 8 -+ 0.06 0.79
k p q x2
0.09 ± 0.17 0.30 -+ 0.07 5.38
1.18 -+ 0.14 1.54 -+ 0.19 - 0 . 1 4 _+ 0.05 1.06
k p q x2
P q x2
0.92 ± 0.08 0.86 ± 0.15 0.05 -+ 0.5 0.88
k p q x2
2
)
5.84 +- 0.39 1.23 -+ 0.05 2.34 ± 0 . 0 6 1.83
5.73 -+ 0.47 1.21 -+ 0.06 2.33 -+ 0.07 1.22
5.45 -+ 0.35 1.18 -+ 0 . 0 4 2.28 -+ 0.05 1.65
1.18 -+ 0 . 0 4 1.34 -+ 0 . 0 4 - 0 . 0 8 ± 0.01 1.70
1.03 -+ 0.04 1.04 _+ 0.06 0.03 ± 0.02 5.71
5.20 -+ 0.39 1.13 -+ 0.05 2.24 +_ 0.06 0.82
1.16_+0.05 1.26 _+ 0 . 0 6 - 0 . 0 4 -+ 0.02 1.43
1.16 ± 0 . 0 4 1.28 _+ 0.05 - 0 . 0 6 _+ 0.01 1,72
1.13 ± 0.04 1.26 -+ 0.05 - 0 . 0 6 -+ 0.01 0.94
6
1.11-+0.06 1.21 _+ 0.08 - 0 . 0 5 -+ 0.02 0.56
1.13 -+ 0.05 1.28 ± 0.07 - 0 . 0 7 -+ 0.02 0.96
1.03 _+0.05 1.06 -+ 0.07 - 0 . 0 1 -+ 0.02 0.74
4
UpT>/On)don/dPTand (]On)don/dPLd i s t r i b u t i o n s for
5.57 ± 0.12 1.19 _+ 0.05 2.31 -+ 0 . 0 6 1.83
5.65 ±- 0.43 1.21 -+ 0.05 2.31 -+ 0.06" 1.20
1.20 -+ 0.04 1.38 +_ 0.05 - 0 . 0 8 ± 0.01 1.82
1.17-+0.05 1.33 ± 0.07 - 0 . 0 7 _+ 0.02 1.21
1.15 ± 0.05 1.25 -+ 0.07 - 0 . 0 4 ± 0.02 2.22
1.09 -+ 0.05 1.17 -+ 0.07 - 0 . 0 3 ± 0.02 1.33
8
5.20 ± 0.43 1.12 -+ 0.05 2.26 -+ 0 . 0 6 1.56
5.31 -+ 0.45 1.12 -+ 0.05 2.28 -+ 0.07 1.27
1.19 -+ 0.05 1.29 -+ 0.07 - 0 . 0 5 -+ 0.02 0.84
1.22-+0.06 1.35 -+ 0.08 - 0 . 0 6 -+ 0.02 0,57
1.16 -+ 0.06 1.25 -+ 0.08 - 0 . 0 4 -+ 0.02 0.97
1.19 -+ 0.06 1.34 ± 0.07 - 0 . 0 7 -+ 0.02 0.96
10+12
d i f f e r e n t n c h at 40 G e V / c
5 . 1 4 _+ 1.30 1.11 -+ 0.17 2.26 ± 0.22 0.52
5.38 -+ 1.35 1.05 -+ 0.16 2.37 _+ 0.22 0.68
1.25 -+ 0.20 1.55 ± 0.28 - 0 . 1 1 ± 0.09 0.57
1.44_+0.17 1.89 -+ 0.20 - 0 . 1 8 ± 0.05 0.91
1.31 -+ 0 . 2 0 1.72 ± 0 . 2 5 - 0 . 1 6 ± 0.06 0.66
1.34 -+ 0.17 1.76 -+ 0.22 0.14 -+ 0.07 1.07
/>14
4+94 -+ 0+17 1.10 -+ 0.02 2 . 2 6 -+ 0,02 2.66
5.32 -+ 0.19 1.16 -+ 0.02 2.26 _+ 0.03 3.24
1.30 -+ 0.02 1.52 ± 0.02 - 0 . 0 1 0 ± 0.01 17.97
1.19±0.03 1.33 ± 0.03 - 0 . 0 7 ± 0.01 0.090
1.23 -+ 0.02 1.41 -+ 0.03 - 0 . 0 9 -+ 0.01 1.54
1.14 +- 0.02 1.27 -+ 0.03 - 0 . 0 7 -+ 0.01 1.99
Inclusive
to
226
N.S. Angelov et al. ~Semi-inclusive cross sections
the longitudinal momentum distributions. Table 4 contains the values of the parameters of the functions (3) and (4) for each multiplicity obtained from the fit. In summary we note the following: (i) The experimental one-particle spectra ((pL)/On) don/dPL and ((pT)/On) don/dPT for different prong multiplicities have a universal character and are reasonably well described b y the functions (3) and (4). (ii) Comparison of our results with the pp data of ref. [1 ] shows that these universal distributions are independent of the type of primary particles, the total energy, and the multiplicity of charged secondaries. It only depends on the scaled momenta
pL / (PL ) and pT / (PT ). The authors express their gratitude to V.G. Grishin for useful discussions and also to the technical staff for their valuable help.
References [1] F.T. Dao et al., Phys. Rev. Letters 33 (1974) 389. [2] A.U. Abdurakhimov et al., Comm. JINR, 1-6326 (Dubna, 1972). [3] A.U. Abdurakhimov et al., Preprint JINR, P1-7267 (Dubna, 1973).
SCALING OF SEMI-INCLUSIVE CROSS SECTIONS WITH RESPECT TO LONGITUDINAL AND TRANSVERSE MOMENTA N.S. ANGELOV, N.N. MELNIKOVA, A.N. SOLOMIN * and L.M. SCHEGLOVA * Joint Institute for Nuclear Research, Dubna N.D. AKHABABYAN, L.D. GREKOVA, V.N. PENEV and A.I. SHKLOVSKAYA Institute for Nuclear Research and Nuclear Energy, Sofia Received 10 April 1975 (Revised 7 July 1975) The semi-inclusiveone-particle transverse and longitudinal momenta spectra of n± mesons have been studied in the centre-of-mass system for 7r-p interactions at 40 GeV/c. These data are compared with similar data from pp interactions at 300 GeV/c and lower energies. Distributions in the variables pT/(PT) and pL/
The analysis of the centre-of-mass inclusive single-particle distributions in the transverse (PT) and the longitudinal (PL) m o m e n t u m variables [1 ] reveals the following scaling property:
1
d2o
1
o dPT dPL = (pT) (PL) f ( P T / ( P T ) ' PL/(PL)) '
(1)
where f is a function, which is independent of multiplicity, incident energy and initial state. This implies that the dynamics are manifest only in the average values (PT) and (PL). In this paper we test this scaling behaviour hypothesis for 7r-p interactions at 40 GeV/c. The data were obtained from the Dubna 2m propane bubble chamber, exposed to a 40 GeV/c rr- beam from the Serpukhov accelerator. We have examined 6275 events of zr-p interactions at 40 GeV/c. These events were selected according with the criteria described in [2]. The one-particle spectra of charged secondary particles for the semi-inclusive process
* Institute for Nuclear Physics Research of Moscow State University.
N.S. Angelov et al. /Semi-inclusive cross sections
220
7r- + p ~ 7r-+ + ( n - - 1) charged particles + neutral particles
(2)
were m e a s u r e d as f u n c t i o n s o f the variables pT/(PT) and pL/(PL). P r o t o n s w e r e i d e n t i f i e d b y i o n i z a t i o n up t o Plab ~ 0.8 G e V / c . All t h e negative particles w e r e t a k e n as pions. In such a p r o c e d u r e the positive particles w e r e c o n t a m i n a t e d b y 5% K + a n d ~ + , and b y ~ 1 6 % n o n - i d e n t i f i e d p r o t o n s [ 3 ] . Table 1 The mean values of longitudinal and transverse momenta for n +- mesons for different topologies in 7r-p interactions at 40 GeV/c nch n
+
2
4
6
8
back forward
0.980 -+ 0.038 0.949 -+ 0.062
0.692 -+ 0.017 0.735 -+ 0.019
0.550 -+ 0.012 0.602 -+ 0.013
0.479 -+ 0.012 0.454 -+ 0.012
back forward
0.493 -+ 0.045 2.006 -+ 0.047
0.555 -+ 0.018 1.121 -+ 0.021
0.503 -+ 0.013 0.746 -+ 0.014
0.410 +- 0.011 0.535 -+ 0.013
7r+ 7r-
0.440 -+ 0.010 0.399 -+ 0.010
0.401 -+ 0.005 0.382 -+ 0.005
0.389 -+ 0.004 0.375 -+ 0.004
0.373 +- 0.004 0.362 -+ 0.005
10 + 12
~> 14
Inclusive
~p~ It
(P±) nch + rr
back forward
0.448 -+ 0.014 0.381 -+ 0.012
0.434 -+ 0.035 0.283 -+ 0.026
0.569 -+ 0.007 0.559 -+ 0.007
~r-
back forward
0.395 +- 0.013 0.423 -+ 0.013
0.684 +- 0.236 0.335 -+ 0.30
0.472 + 0.011 0.838 -+ 0.009
0.363 -+ 0.006 0.341 + 0.005
0.324 +- 0.015 0.301 -+ 0.012
0.384 -+ 0.002 0.367 -+ 0.002
+
(P±)
~r 7r-
Table 1 s h o w s the average values o f transverse and l o n g i t u d i n a l m o m e n t a in c.m.s. for e a c h m u l t i p l i c i t y o f charged s e c o n d a r y particles in n - p i n t e r a c t i o n s . These values have b e e n o b t a i n e d b y averaging the transverse and longitudinal m o m e n t a w i t h i n t h e k i n e m a t i c ranges d e t e r m i n e d for e a c h m u l t i p l i c i t y , taking into a c c o u n t t h e measurem e n t errors. Fig. 1 (a, b) illustrates t h e ((pT)/On) don/dPT shapes for d i f f e r e n t c h a n n e l s o f r e a c t i o n (2) for n - and n + m e s o n s , respectively. One can observe a universal behaviour o f t h e s e d i s t r i b u t i o n s , w h i c h are i n d e p e n d e n t o f nch. The d a t a w e r e fitted in c o m m o n for d i f f e r e n t nch t o t h e f u n c t i o n [ PT ~ c
(3)
A
(a)
.':
"''i:.
a5
Pl
65
o .
.~-
(b)
PJ. ~p.~> ~,+ Fig. 1. Plot o f ((pT>/On) dan~alp T versus p T / ( P T ) for t h e r e a c t i o n s 7r-p ~ 7r- (a) and 7r-p ~ 7r+ (b) for d i f f e r e n t n c h at 40 GeV/c. T h e solid curve p r e s e n t s t h e fit o f 7r-p d a t a at 40 G e V / e . The d a s h e d curve is t h e fit results o f t h e e x p e r i m e n t a l p p d a t a [1 ]. Table 2 T h e p a r a m e t e r s o f t h e universal f u n c t i o n s (3) and (4), d e s c r i b i n g in c o m m o n the 7r+ m e s o n distributions: (PT) b~ *
*
do n dp T '
(PL) d a n
on
dp L
k
p
q
x2/(N-M)
1.14 + - 0.03
1.27 + - 0.04
-0.05 + - 0.01
1.14
7¥
1.18 -+ 0 . 0 4
1.34 -+ 0.05
- 0 . 0 7 -+ 0.01
1.31
nB
1.20 + 0.04
1.36 + 0 . 0 4
- 0 . 0 7 -+ 0.01
1.37
7rF
1.17 + 0 . 0 4
1.31 +- 0 . 0 4
- 0 . 0 6 + 0.01
1.89
7rBa)
0.91 + 0.15
0.83 + 0 . 0 4
0.03 +-0.01
0.96
p.L/(p±)
a
c
b
x2/(N-M)
~+ ~r-
5.41 -+ 0.29 5.21 +- 0.27
1.14 _+ 0 . 0 4 1.14 -+ 0 . 0 4
2.28 -+ 0 . 0 4 2.25 -+ 0 . 0 4
1.06 1.25
~r-
6.23 -+ 0.52
1.37 +- 0.03
2.37 -+ 0.04
1.34
PlI/(PII ) 7r + B +
a) p p d a t a , see ref. [ 1 ] .
~ b ¢.0 09" 08" 0.7@~0.5" at"
t
a2 0A
s
"~,, •
~5
¢.5
i~k
.
2,5
35
~5 < p,*>
~05
" "~'
t
O.t °. ,
O.5
~5
#,5
,:15
45 II.
O8
g7
O5" 0~03-
0.~
t
0~
~
~
.15
i
I
o,,"
~5
f.t
Qg
a7"
"~
0.5
~x
0/,/
•
~,\
03
OZ Ill i
;
Fig. 3. Plot of ((pL)/On) don/dPL versus pL/(PL) for the reaction 7r-p ~ I t - in the regions PL > 0 (~F-) and PL < 0 (~B-) for different rich at 40 GeV/c. The dashed curve is taken from ref. [11.
N.S. Angelov et al. /Semi-inclusive cross sections
224
Table 3 The values of mean square deviations of the universal functions (3) and (4) with fixed parameters (see table 2) from the 40 GeV/c data at different nch nch
P±
2
4
6
8
10,12
/>14
Inclusive
7r-t-
back forward
1.19 1.20
1.10 1.09
1.15 1.79
1.41 2.47
1.01 1.15
1.63 0.97
2.54 2.07
7r-
back forward
1.01 7.75
0.76 8.66
1.52 2.20
1.33 1.90
0.60 0.92
1.31 0.61
1.45 20.76
1.16 1.51
1.29 0.83
2.07 1.70
1.38 1.86
1.30 1.60
0.86 0.53
3.57 2.76
~r+ n
and the universal parameters a, c and b were obtained (see table 2). For comparison the corresponding universal parameters for pp interactions [1 ] are also given in table 2. The dashed curve in fig. 1 corresponds to the data o f ref. [1 ]. The solid curve is obtained by fitting the data from 7r-p interactions at 40 GeV/c. As is seen from fig. 1, b o t h curves fit our data well within the experimental errors. In order to check to what extent our data is described b y the universal functions with the parameters obtained b y fitting the semi-inclusive data in common, we calculated the mean square deviations o f these functions from experimental distributions at different rich. The result is presented in table 3. It is seen that ((pT)/On) dan/dPw distributions are well described b y the universal function (3). To analyse the scaling properties o f the longitudinal momentum distributions, we have plotted the spectra ((pL)/On) don/dPL for the regions PL > 0 and PL < 0 separately. These spectra are given in figs. 2(a, b) and 3(a, b) for 7r+ and n - mesons respectively. The universal functions obtained by fitting the ~r- p data at 40 GeV/c (solid curve) and the universal function from paper [1 ] (dashed curve) are also shown in figs. 2 and 3. The experimental distributions for different rich were fitted in common to the following function
tpL = k exp
p
P((~L)) [ PL ]2] - qt(-~L)j J.
(4)
The parameters k, p and q are given in table 2. One can see from figs. 2 and 3, as well as from table 3, that the experimental data on 71.+ (sOL <~ 0) and 7r- (PL < 0) are well described b y the function (4) within the experimental errors. Some deviations from the universal behaviour come from the events with low multiplicity o f charged secondary particles or due to large measurement errors. The distributions for 7r- (PL > 0) (see fig. 3) are dominated b y a contribution from leading 7r- mesons for which one should not expect the scaling behaviour o f
P±
forward
zr-
7r +
back
forward
back
7r-
~r
+
~+
nch
Table 4 The results of the fit of
a c b x2
4.02 -+ 0.55 0 . 9 4 _+ 0.09 2.07 -+ 0.11 1.33
6,57 -+ 1.18 1.35 _+ 0.13 2.45 _+ 0.15 1.02
0.60 ± 0.06
k
a c b x2
1.38_+0.21 1.84 _+ 0.25 - 0 . 1 8 -+ 0.06 0.79
k p q x2
0.09 ± 0.17 0.30 -+ 0.07 5.38
1.18 -+ 0.14 1.54 -+ 0.19 - 0 . 1 4 _+ 0.05 1.06
k p q x2
P q x2
0.92 ± 0.08 0.86 ± 0.15 0.05 -+ 0.5 0.88
k p q x2
2
)
5.84 +- 0.39 1.23 -+ 0.05 2.34 ± 0 . 0 6 1.83
5.73 -+ 0.47 1.21 -+ 0.06 2.33 -+ 0.07 1.22
5.45 -+ 0.35 1.18 -+ 0 . 0 4 2.28 -+ 0.05 1.65
1.18 -+ 0 . 0 4 1.34 -+ 0 . 0 4 - 0 . 0 8 ± 0.01 1.70
1.03 -+ 0.04 1.04 _+ 0.06 0.03 ± 0.02 5.71
5.20 -+ 0.39 1.13 -+ 0.05 2.24 +_ 0.06 0.82
1.16_+0.05 1.26 _+ 0 . 0 6 - 0 . 0 4 -+ 0.02 1.43
1.16 ± 0 . 0 4 1.28 _+ 0.05 - 0 . 0 6 _+ 0.01 1,72
1.13 ± 0.04 1.26 -+ 0.05 - 0 . 0 6 -+ 0.01 0.94
6
1.11-+0.06 1.21 _+ 0.08 - 0 . 0 5 -+ 0.02 0.56
1.13 -+ 0.05 1.28 ± 0.07 - 0 . 0 7 -+ 0.02 0.96
1.03 _+0.05 1.06 -+ 0.07 - 0 . 0 1 -+ 0.02 0.74
4
UpT>/On)don/dPTand (
5.57 ± 0.12 1.19 _+ 0.05 2.31 -+ 0 . 0 6 1.83
5.65 ±- 0.43 1.21 -+ 0.05 2.31 -+ 0.06" 1.20
1.20 -+ 0.04 1.38 +_ 0.05 - 0 . 0 8 ± 0.01 1.82
1.17-+0.05 1.33 ± 0.07 - 0 . 0 7 _+ 0.02 1.21
1.15 ± 0.05 1.25 -+ 0.07 - 0 . 0 4 ± 0.02 2.22
1.09 -+ 0.05 1.17 -+ 0.07 - 0 . 0 3 ± 0.02 1.33
8
5.20 ± 0.43 1.12 -+ 0.05 2.26 -+ 0 . 0 6 1.56
5.31 -+ 0.45 1.12 -+ 0.05 2.28 -+ 0.07 1.27
1.19 -+ 0.05 1.29 -+ 0.07 - 0 . 0 5 -+ 0.02 0.84
1.22-+0.06 1.35 -+ 0.08 - 0 . 0 6 -+ 0.02 0,57
1.16 -+ 0.06 1.25 -+ 0.08 - 0 . 0 4 -+ 0.02 0.97
1.19 -+ 0.06 1.34 ± 0.07 - 0 . 0 7 -+ 0.02 0.96
10+12
d i f f e r e n t n c h at 40 G e V / c
5 . 1 4 _+ 1.30 1.11 -+ 0.17 2.26 ± 0.22 0.52
5.38 -+ 1.35 1.05 -+ 0.16 2.37 _+ 0.22 0.68
1.25 -+ 0.20 1.55 ± 0.28 - 0 . 1 1 ± 0.09 0.57
1.44_+0.17 1.89 -+ 0.20 - 0 . 1 8 ± 0.05 0.91
1.31 -+ 0 . 2 0 1.72 ± 0 . 2 5 - 0 . 1 6 ± 0.06 0.66
1.34 -+ 0.17 1.76 -+ 0.22 0.14 -+ 0.07 1.07
/>14
4+94 -+ 0+17 1.10 -+ 0.02 2 . 2 6 -+ 0,02 2.66
5.32 -+ 0.19 1.16 -+ 0.02 2.26 _+ 0.03 3.24
1.30 -+ 0.02 1.52 ± 0.02 - 0 . 0 1 0 ± 0.01 17.97
1.19±0.03 1.33 ± 0.03 - 0 . 0 7 ± 0.01 0.090
1.23 -+ 0.02 1.41 -+ 0.03 - 0 . 0 9 -+ 0.01 1.54
1.14 +- 0.02 1.27 -+ 0.03 - 0 . 0 7 -+ 0.01 1.99
Inclusive
to
226
N.S. Angelov et al. ~Semi-inclusive cross sections
the longitudinal momentum distributions. Table 4 contains the values of the parameters of the functions (3) and (4) for each multiplicity obtained from the fit. In summary we note the following: (i) The experimental one-particle spectra ((pL)/On) don/dPL and ((pT)/On) don/dPT for different prong multiplicities have a universal character and are reasonably well described b y the functions (3) and (4). (ii) Comparison of our results with the pp data of ref. [1 ] shows that these universal distributions are independent of the type of primary particles, the total energy, and the multiplicity of charged secondaries. It only depends on the scaled momenta
pL / (PL ) and pT / (PT ). The authors express their gratitude to V.G. Grishin for useful discussions and also to the technical staff for their valuable help.
References [1] F.T. Dao et al., Phys. Rev. Letters 33 (1974) 389. [2] A.U. Abdurakhimov et al., Comm. JINR, 1-6326 (Dubna, 1972). [3] A.U. Abdurakhimov et al., Preprint JINR, P1-7267 (Dubna, 1973).