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Energy dependence of topological cross-sections at nal energies
Volume 56B, n u m b e r 3
PHYSICS LETTERS
ENERGY
DEPENDENCE
CROSS-SECTIONS
28 April 1975
OF TOPOLOGICAL
AT NAL ENERGIES
J. GANDSMAN Universit~ Paris-Sud, Laboratoire de l'Acc~ldrateur Lindaire, 91405 Orray, France Received 20 January 1975 Revised manuscript received 3 March 1975 We calculate the normalized cross-sections o* = a ' FLUX/LIPS using recent data on topological cross-sections.
We confirm the power law behaviour a* = APla~ for energies up to 400 GeV. The exponent n appears to satisfy, approximately, the relation n ~ m - 2 for multiplicities m up to 20.
The cross-section for a reaction, where N particles are produced in the final state, may be written as: oN = ISI2v • LIPS/FLUX.
(1)
LIPS stands for the integrated Lorentz invariant phase space ¢ and the FLUX is defined as FLUX = P c m ~ with Pcm = center o f mass momentum and S = square of the center o f mass energy, tSla2vrepresents the dynamical factor and is the square o f the transition matrix elements averaged over the whole phase space volume. It is reasonable, thus, to study the energy dependence o f cross-sections normalized to a constant phase space. This was first done by Muirhead and Poppleton [1 ] for pp annihilation and subsequently by Hofmokl and Wroblewski [2] for a variety of reactions. The normalized cross-sections o* defined as FLUX o* = o" LIP----S- ISla2v
(2)
ble, it is interesting to review this behaviour o f inelastic cross-sections over the new domain o f energies. This could also allow a precise determination o f the functional dependence o f n on the multiplicity. We use for our analysis the compilation o f inelastic topological cross-sections presented by Whitmore (references therein) which are shown in fig. 1. They all follow the well known pattern: rise from threshold to a broad maximum and then decrease with the increasing momentum of the beam. We calculate a* according to expression ~ (2) and the results are presented in figs. 2, 3 and 4 for pp, rr+p and 7r-p reactions, respectively. All the a* cross-sections are well described by the power law formula and in our log-log graphs straight lines fit the data very well in all the energy range and for all the multiplicities. Next, we study the dependence o f n on the multiplicity and/or the mass o f the beam particle. In fig. 5 we show the computed values of n, for all the reactions, as a function o f the multiplicity rn. The functional dependence is very well described by the expressions
had an energy dependence well described b y a simple power law
pp
o* = A P ~ .
lr±p : n -- (0.971 -+0.06)m - (2.30-+0.05) (X 2 = 1.5).
These analyses covered an energy range between 1 - 2 0 GeV and multiplicities up to 8. The value o f n seemed to be an increasing function of the multiplicities [2]. Today, when data from the new accelerators is availa-
As we can see there is a very small dependence on the
* As defined in ref. [21.
286
: n = (0.991 -+0.06) m - (2.22-+0.04) (X 2 = 5.9)
* It is important to remark that the LIPS factor is approxim a t e d because we have not considered the neutral contribu-
tion. For example, for the 4 prongs pp and np topological cross-sections we calculate the LIPS of pp --* ppn~r and Trp ~ 7rp~rrr(and similarly for the others).
Volume 56B, number 3
PHYSICS LETTERS
~!~.
L I
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28 April 1975
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287
pp Topological cross-sections (o') !
101
- - "
~
f--
f--
I
< ~,
~+p Topological cross-sections(o")
101 .
4
i
1 0 -3
1 0 -1
10 -7
I
"b
10-3
1
10-11 10-5
i0-I"16
+
10-7
10 -le 10-9
'
2'0
'
'do''
J , 1
lOO
2~o
'~
460
Plab(GeV/c)
--.
i
2'0
55
16o 200
P I=b(GeV/c)
Fig. 2. o* topological cross-sections for pp reactions. p
1'o
i
Fig. 3. e* topological cross-sections for ~r+p reactions. i
i
"p. Topological cross-sections((r')
20
i
i
I
#
I
#
I
i
1 0 -1
• pp • I%-p
t
w II÷p
10-3
I
n
#
15
+
10-5
"b 10 10-7
10-9
T
I0-111 10-131 1'0
2=
16 t
20O 510 1C)0 Plib(GeV/c) Fig. 4. o* topological cross-sections for ~-p reactions.
288
2'0
I
4
8
12
Multiplicity (m)
16
20
Fig. 5. Computed values of the exponent n as a function of the multiplicity.
Volume 56B, number 3
PHYSICS LETTERS
28 April 1975
type of incident particle (but detectable) and seems to follow, approximately, the law • lt-p ,, lt+p 4 P r o n g s +~t*p 6 P r o n g s
10 °
I
• pp
I Jf
n~m-2.
1
The 7r+p and ~ - p data essentially coincide and if we disregard the slightly larger slope of pp reactions, the dependence on the mass of the projectile is contained in the parameter A of formula (3). This can also be seen in fig. 6 where we plot in the same graph, for comparison, the 4 and 6 prongs o* cross-sections for ~ttp and pp reactions. Formula (3) may be considered a generalization of the law found by Morrison [4, 5] in the study of two body reactions. The energy dependence of two body reactions was in agreement with the law o = AP~a~. As it was pointed out by Hofmokl and Wroblewski, for sufficiently high energy, the two body phase space volume is constant and the only difference between formula (3) and Morrison's law is the flux factor which at high energies is proportional to Plab" Independently of any theoretical consideration formula (3) gives a convenient parametrization of inelastic cross-sections and it can be used also to predict the values of cross-sections at higher energies. As an example we show in fig. 7 the predictions for the 4 - 6 8 prongs topological cross-sections for energies up to 104 GeV.
10--2.
% 10-3
10- 4
10 - 5
,o-'l
~
lb
2'o
sb l o' p 2oo ' Plab (GeV/c)
's"o p
loop
Fig. 6. o* distribution for 4 and 6 prongs in ~r±p and pp reactions.
Topological cross-sections 1~-o e ,
*'"~-A -A.. 10
N ..... "~y'*-t-tw~ ,
n(m)
p ..
,A= 4 68
I
Prongs
~''A.
I
-4~p
4 t
10
IO ~
10 3
Phlb (GeV/c)
Fig. 7. Topological cross-sections for 4 - 6 - 8 prongs. The full curves represent our predictions for higher energies. Again the dotted curves are only drawn to guide the eye.
289
Volume 56B, number 3
PHYSICS LETTERS
References [l] H. Muirhead and A. Poppleton, Phys. Lett. 29B (1969) 448.
290
28 April 1975
[2] T. Hofmokl and A. Wroblewski, Phys. Lett. 31B (1970) 391. [3] J. Whitmore, Phys. Reports 10C (1974) 273. [4] D.R.O. Morrison, Phys. Lett. 22 (1966) 528. [5] D.R.O. Morrison, Phys. Rev. 165 (1968) 1699.
PHYSICS LETTERS
ENERGY
DEPENDENCE
CROSS-SECTIONS
28 April 1975
OF TOPOLOGICAL
AT NAL ENERGIES
J. GANDSMAN Universit~ Paris-Sud, Laboratoire de l'Acc~ldrateur Lindaire, 91405 Orray, France Received 20 January 1975 Revised manuscript received 3 March 1975 We calculate the normalized cross-sections o* = a ' FLUX/LIPS using recent data on topological cross-sections.
We confirm the power law behaviour a* = APla~ for energies up to 400 GeV. The exponent n appears to satisfy, approximately, the relation n ~ m - 2 for multiplicities m up to 20.
The cross-section for a reaction, where N particles are produced in the final state, may be written as: oN = ISI2v • LIPS/FLUX.
(1)
LIPS stands for the integrated Lorentz invariant phase space ¢ and the FLUX is defined as FLUX = P c m ~ with Pcm = center o f mass momentum and S = square of the center o f mass energy, tSla2vrepresents the dynamical factor and is the square o f the transition matrix elements averaged over the whole phase space volume. It is reasonable, thus, to study the energy dependence o f cross-sections normalized to a constant phase space. This was first done by Muirhead and Poppleton [1 ] for pp annihilation and subsequently by Hofmokl and Wroblewski [2] for a variety of reactions. The normalized cross-sections o* defined as FLUX o* = o" LIP----S- ISla2v
(2)
ble, it is interesting to review this behaviour o f inelastic cross-sections over the new domain o f energies. This could also allow a precise determination o f the functional dependence o f n on the multiplicity. We use for our analysis the compilation o f inelastic topological cross-sections presented by Whitmore (references therein) which are shown in fig. 1. They all follow the well known pattern: rise from threshold to a broad maximum and then decrease with the increasing momentum of the beam. We calculate a* according to expression ~ (2) and the results are presented in figs. 2, 3 and 4 for pp, rr+p and 7r-p reactions, respectively. All the a* cross-sections are well described by the power law formula and in our log-log graphs straight lines fit the data very well in all the energy range and for all the multiplicities. Next, we study the dependence o f n on the multiplicity and/or the mass o f the beam particle. In fig. 5 we show the computed values of n, for all the reactions, as a function o f the multiplicity rn. The functional dependence is very well described by the expressions
had an energy dependence well described b y a simple power law
pp
o* = A P ~ .
lr±p : n -- (0.971 -+0.06)m - (2.30-+0.05) (X 2 = 1.5).
These analyses covered an energy range between 1 - 2 0 GeV and multiplicities up to 8. The value o f n seemed to be an increasing function of the multiplicities [2]. Today, when data from the new accelerators is availa-
As we can see there is a very small dependence on the
* As defined in ref. [21.
286
: n = (0.991 -+0.06) m - (2.22-+0.04) (X 2 = 5.9)
* It is important to remark that the LIPS factor is approxim a t e d because we have not considered the neutral contribu-
tion. For example, for the 4 prongs pp and np topological cross-sections we calculate the LIPS of pp --* ppn~r and Trp ~ 7rp~rrr(and similarly for the others).
Volume 56B, number 3
PHYSICS LETTERS
~!~.
L I
~IG? Ib
%
..~ ~I II
~§
/:
~,
t k'~
\
~.
"+. • , ,
"...
28 April 1975
.,...
:,--¢-....
-....
'"-
--...
-
"
. . . . . . . . .
@
/
~k
*,.
,,, n..
%,.
I
~.
.~.,,... "-,,.
.,04
:
.,.
-...~=_
o ...... @
,o ....
2- - . . . . . . . .
_'"~
~|
x,~+
8
,y
. . . .
"*r" t
_t . . . . . . . .
_, . . . . . . . .
•
o.
-o-
~
.
e. 0
.,
i-
lu
II
•
,-,
~,.
~'%
P
'~.....
~
.~
'~,~
-~-.
_
"2_
" , .,.
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9 I .-J li*
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I~
~ /,
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i O-,.
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,
,
,
i
|.,w
r
r
i
r
i
].lu
|
I
l
l
1
0
287
pp Topological cross-sections (o') !
101
- - "
~
f--
f--
I
< ~,
~+p Topological cross-sections(o")
101 .
4
i
1 0 -3
1 0 -1
10 -7
I
"b
10-3
1
10-11 10-5
i0-I"16
+
10-7
10 -le 10-9
'
2'0
'
'do''
J , 1
lOO
2~o
'~
460
Plab(GeV/c)
--.
i
2'0
55
16o 200
P I=b(GeV/c)
Fig. 2. o* topological cross-sections for pp reactions. p
1'o
i
Fig. 3. e* topological cross-sections for ~r+p reactions. i
i
"p. Topological cross-sections((r')
20
i
i
I
#
I
#
I
i
1 0 -1
• pp • I%-p
t
w II÷p
10-3
I
n
#
15
+
10-5
"b 10 10-7
10-9
T
I0-111 10-131 1'0
2=
16 t
20O 510 1C)0 Plib(GeV/c) Fig. 4. o* topological cross-sections for ~-p reactions.
288
2'0
I
4
8
12
Multiplicity (m)
16
20
Fig. 5. Computed values of the exponent n as a function of the multiplicity.
Volume 56B, number 3
PHYSICS LETTERS
28 April 1975
type of incident particle (but detectable) and seems to follow, approximately, the law • lt-p ,, lt+p 4 P r o n g s +~t*p 6 P r o n g s
10 °
I
• pp
I Jf
n~m-2.
1
The 7r+p and ~ - p data essentially coincide and if we disregard the slightly larger slope of pp reactions, the dependence on the mass of the projectile is contained in the parameter A of formula (3). This can also be seen in fig. 6 where we plot in the same graph, for comparison, the 4 and 6 prongs o* cross-sections for ~ttp and pp reactions. Formula (3) may be considered a generalization of the law found by Morrison [4, 5] in the study of two body reactions. The energy dependence of two body reactions was in agreement with the law o = AP~a~. As it was pointed out by Hofmokl and Wroblewski, for sufficiently high energy, the two body phase space volume is constant and the only difference between formula (3) and Morrison's law is the flux factor which at high energies is proportional to Plab" Independently of any theoretical consideration formula (3) gives a convenient parametrization of inelastic cross-sections and it can be used also to predict the values of cross-sections at higher energies. As an example we show in fig. 7 the predictions for the 4 - 6 8 prongs topological cross-sections for energies up to 104 GeV.
10--2.
% 10-3
10- 4
10 - 5
,o-'l
~
lb
2'o
sb l o' p 2oo ' Plab (GeV/c)
's"o p
loop
Fig. 6. o* distribution for 4 and 6 prongs in ~r±p and pp reactions.
Topological cross-sections 1~-o e ,
*'"~-A -A.. 10
N ..... "~y'*-t-tw~ ,
n(m)
p ..
,A= 4 68
I
Prongs
~''A.
I
-4~p
4 t
10
IO ~
10 3
Phlb (GeV/c)
Fig. 7. Topological cross-sections for 4 - 6 - 8 prongs. The full curves represent our predictions for higher energies. Again the dotted curves are only drawn to guide the eye.
289
Volume 56B, number 3
PHYSICS LETTERS
References [l] H. Muirhead and A. Poppleton, Phys. Lett. 29B (1969) 448.
290
28 April 1975
[2] T. Hofmokl and A. Wroblewski, Phys. Lett. 31B (1970) 391. [3] J. Whitmore, Phys. Reports 10C (1974) 273. [4] D.R.O. Morrison, Phys. Lett. 22 (1966) 528. [5] D.R.O. Morrison, Phys. Rev. 165 (1968) 1699.