Recent theoretical investigations of high energy nucleon-nucleon collisions support the two-center model of multiple meson production as discussed by Ciok, et al. To facilitate a prediction of the most probable distribution of shower particles as a function of laboratory energy, Farley's kinematical treatment of the "two-fireball" model is

employed. A statistical determination of the multiplicity of pion (including neutrals) production per collision is assumed to be valid with possible corrections suggested through restrictions imposed by experimental values of the inelasticity, Κ_π

Hankins, Denis

Watson, Keith R.

Caltech Authors - Main



Report CTSL-29
CALIFORNIA INSTITUTE OF TECHNOLOGY
Synchrotron Laboratory
Pasadena, California
PION PRODUCTION IN 300 GEV NUCLEON-NUCLEON COLLISIONS
Keith R. Watson* 
and
Denis Hankins*
June 5, 1961
*University of Southern California
Supported in part by the U.S. Atomic Energy Commission 
Contract No. Α Τ ( 11 - 1 )-68
1Contents
I Introduction p. 2
II Development p. 4
III Laboratory System Shower Distribution p . 6
IV Effect of Inelasticity on n s p. 10
Appendix p . 12
Abstract
Recent theoretical investigations of high energy nucleon-nucleon
collisions1) support the two-center model of multiple meson production as
discussed by Ciok, et al.2 ) . To facilitate a prediction of the most
probable distribution of shower particles as a  function of laboratory
energy, Farley's kinematical treatment of the "two-fireball" model is
employed 3). A  statistical determination of the multiplicity of pion 
(including neutrals) production per collision is assumed to be valid with 
possible corrections suggested through restrictions imposed by experi­
mental values of the inelasticity, Κπ .
Acknowledgments
The authors would like to take this opportunity to thank 
Professors Matthew Sands and Harold Ticho for making several interesting 
comments and for participating in many useful discussions.
2I Introduction
A n  analysis of multiple pion production induced by nucleon-nucleon 
collisions is developed on the basis of a two fireball statistical model. 
At 300 Gev primary energy, the statistical model based purely upon phase 
space arguments indicates that about six pions per fireball are created 
per collision. Experimental evidence however, shows that the mean pion 
energy is essentially almost independent of excitation energy (i.e., 0.5 
Gev) in the excited nucleon system. This coupled with an average number 
of six pions (multiplicity of 12), predicts an inelasticity about twice 
as large as that observed experimentally. This suggests, therefore, that 
the number of pions produced per collision (on a statistical basis) should 
be modified b y  matrix element considerations.
Experimental cosmic ray data appears to suggest that the value for 
the inelasticity is in the neighborhood of 0.3 to 0.4 at 300 Gev lab
energy4 ) , at the same time supporting the two fireball concept. In order 
for this value to agree with this model, it is necessary to reduce either 
ns (multiplicity) or the average pion energy in the excited nucleon sys-
tem Ē . Some experiments favor values of Ē as low as 0.4, instead of the
usually accepted value ~ 0 . 5 4). However, this reduction in Ē is still 
not sufficient to lower Kπ to the "correct" value. It is apparent th e r e ­
-fore that ns must be lowered in order to obtain better agreement. Pure 
phase space and statistical equilibrium arguments (a la Fermi) yields a 
value ~ 12 pions/collision (i.e., 6 pions per fireball), hence matrix 
element, or non-equilibrium considerations must be employed in order to 
lower this value.
3More recent w o r k  based upon "The Eightfold Way"5) has shown that 
the elastic scattering cross section approaches a constant value of 
approximately 40 millibarns at infinite energy1 ) . In addition, experi­
-me n t a l  cosmic ray data also suggests that the inelastic cross section
approaches a constant value in the vicinity of 20 millibarns4). This
indicates, therefore, that the probability per collision of creating
"vector mesons" is about one-third. If we make the further assumption
that single vector meson production is more probable than multiple vector
meson production* , we should expect approximately one pion to be produced
per collision. (This arises from decays such as ρ  →  2π, ω  → 3π.) By
making the further plausible assumption that the average pion energy in
the excited nucleon system is almost constant ~ 0.5 Gev, we may arrive at
an asymptotic value of the inelasticity 0.2, which is in good agreement
with the high energy experimental results6 ) ·
A  direct consequence of lowering the value of n s is to lower the 
excitation energy of the fireballs. This automatically increases the 
value of ϒ , and hence pushes the narrow cone pion distribution to higher 
energies. This is apparently undesirable since some experiments indicate 
that the pion energy spectrum cuts off at energies lower than that pr e ­
-dicted by the Farley model with ns ~ 12. At the present time, there
appears to be no wa y  of resolving this paradox* * .
* This is supported by  recent experiments at 4 Gev; however, this energy 
is probably not large enough to guarantee this statement.
*One possibility is currently being investigated by H. Ticho.
4In the development, Farley's model wi l l  be exploited in an attempt 
to obtain approximate energy and angular distributions of pions in the 
laboratory.
II Development
We shall consider an inelastic collision of an incoming primary 
nucleon (described b y  energy E p  and Lorentz factor ϒ p ) wi t h  a nucleon 
stationary in the laboratory system (LS). Let βc denote the velocity of 
the center of mass system (CS) where
(1)The two nucleons of rest mass m o and total energy ϒc m o = Ec
approach each other w i t h  equal velocities βc in the CS. After collision,
the mass of each nucleon is increased to ϒ-1 Ec = m o* the velocities of
the two excited nucleons being reduced to β. The energy relation thus 
becomes:
(2)Assuming n s /2 pions per nucleon are emitted isotropically with
average energy Ē 3) in the excited nucleon rest system (N S ) , the excita­
-tion energy m a y  be written in the form
(3)
FIG. 1 KINEMATICS AFTER COLLISION
6If we let ϒ f and ϒ b describe the motion in the LS of the excited 
nucleons which are emitted in the same and the opposite direction to βp 
in the CS (Fig. 1 ) , it can be easily shown that
(4)
(5)
I LabortySsemhwDiun:(Ep = 300 Gev)
For Ep = 300 Gev, a  Fermi statistical analysis of the meson cloud
yields n s /2 =  6. Thus the E x =  3 curve of Farley's3) energy spectrum of
mesons in the NS ma y  be chosen as that wh i c h  wi l l  yield, under transfor- 
mation, the most probable energy distribution of shower particles in the 
LS. (See Fig. 2.)
The following relations and values will b e  important. From (1 )
(6)
from (3), (2),
(7)
and from (4), (5),
(8)
(9)
Choosing points from Fig. 2 and applying the relation
(10)
RELATIVE PROBABILITY, P (Eπ ) N OF MESON 
EMISSION AS A FUNCTION OF THEIR ENERGIES.
FIG. 2.
8the following table is obtained:
P(Eπ )N 0.2 0.38 0.42 0.36 0.12 0.02
(Επ )N 0.15 Gev 0.25 0.50 0.565 (mean)
1.0 1.5
(Ρπ )Ν 0.07 Gev 0.21 0.49 0.55 0.99 1.5
TABLE 1
Note that the m e a n  pion energy is in good agreement (in the r e l a ­
-tivistic limit), wi t h  the experimental value of the mean transverse 
momentum PT  = 0.5 Gev/c obtained by Edwards, et al.7), and others.
At 300 Gev, the majority of pions, w h e n  transformed b y  the r e l a ­
tions
(1)and
(12)
will be concentrated in a 1-3 degree cone in the forward direction (see 
A p p e n d i x ) . The energy distribution associated wi t h  the forward cone is 
shown in Fig. 3. Since the backwards cone occupies only a small part 
(0-10 Gev) of the (LS) transformed energy spectrum, Fig. 3 contains 
essentially all the information of interest. The spectrum was transformed 
in the following straightforward manner. Each of the five points in 
Table 1 was transformed according to (1 1 ) for 15° intervals ranging from 
0 to 180 degrees. Since NS emission is isotropic, P(Eπ )N is proportional 
to the solid angle of emission, and therefore weighted as sin θΝ · The
FIG. 3
10
constant θ N curves were added and the resultant curve was renormalized to 
1/2 n s = 6.
IV Effect of Inelasticity on  n s
The inelasticity, Kπ , in a nucleon-nucleon collision, is defined
as that fraction of the total energy available in the CS which is sub-
sequently evaporated in the form of pions. F r o m  Ciok, et al.2 ) , we have
(13)from (3)
hence
and finally
(14)For our particular case n s = 12, Ē ≈  0.56 5, thus Kπ = 0.78.
Gierula, Haskin, and Lohrmann8 ) , and others find values of the
inelasticity which tend towards 0.3 in this energy range. GH L  have used
a low multiplicity, ns =  4, two-center model which closely predicts LS
angular distributions. This work, coupled w i t h  later studies of the 
Bristol experiments by Perkins, seems to demand a re-evaluation of n s ·
A statistical mo d e l  in which only small-angle scattering is impor- 
tant will tend to lower the inelasticity somewhat, since the "core" of
11
the nucleon w i l l  not be excited. Recent w o r k  by Gell-Mann5) has indicated 
that vector mesons may play a strong role in nucleon-nucleon interactions.
On the basis of our extension of this work, it has been shown4) that
(dσ/dΩ)e l a s t i c  is strongly peaked in the forward direction at high energies,
the same behavior is also indicated for the contribution to (dσ/dΩ)inelastic.
This, therefore, suggests that the momentum transfer at high energies is
fairly small, thus reducing the effective excitation energy, and hence
the value of ns and K π derived solely on the basis of the statistical
model. (This again causes an increase in ϒ .)
In conclusion, it should be explained that the two fireball model
of N-N collisions is probably correct for all energies. Sakurai9) has 
suggested that the repulsive nature of the nucleon core should give rise 
to two excited nucleon fireball systems. However, the approximation 
employed in the Fermi statistical model to these separate fireball sys­
-tems is probably invalid since the excitation energy w i l l  tend to remain 
small, even at high energies1 ) . This would probably not be the case for 
N-N systems where a  strong attractive potential exists (possibly even 
large enough to give bound states (e.g., π meson)1 0 ) ), this process would 
clearly enhance the tendency towards a single fireball description, and 
it is hoped that this point can ultimately be checked experimentally.
12
APPENDIX
Number of pions as a function of L S  angle, θ L . For
(1A)Let 1/2 n s be the number of  pions per fireball emitted isotropically 
in the NS. By conservation of particles
(2A)differentiating (1A) and combining wi t h  (2A) gives
o r
(3A)similarly
(4A)Integrai (3)d4
Hence, one may conclude that there are about three pions per col- 
lision in the first degree cone and six in the three degree forward cone, 
on the assumption of six pions per fireball as derived on the basis of 
the statistical model.
13
REFERENCES
1. W. K. R. Watson, B u l . Am. Phys. Soc. II, 6, 372 (1961 ) .
2. P. Ciok, T. Coghen, J. Gierula, R. Holynski, A. Jurak, M. Miesowicz,
T. Saniewska, 0. Stanisz, and J. Pernegr, Nuovo Cimento 8, 166 (1958).
3. F. J. M. Farley, Nuovo Cimento 16, 209 (1 960).
4. D. H. Perkins, Elementary Particle and Cosmic Ra y  Physics, V, (1 960), 
No. Holland Pub. Co., Amsterdam, pp. 259-360.
5. M. Gell-Mann, (to be published in Phys. Rev.) 1961 .
6. G. Bertolino, Nuovo Cimento 2, 1130 (1955).
7. B. Edwards, J. Losty, K. Pinkau, D. H. Perkins, and J. Reynolds, 
Phil. Mag. 3, 237 (1958).
8. J. Gierula, D. M. Haskin, and E. Lohrmann, Phys. Rev. 122, 626 (1 961 ) .
9. J. J. Sakurai, Ann. Phys. (Sept. 1960).
10. P. K a u s , et a l . , B u l . Am. Phys. Soc. II, 6, 377 (1961 ).




Pion Production in 300 Gev Nucleon-Nucleon Collisions



Report CTSL-29
CALIFORNIA INSTITUTE OF TECHNOLOGY
Synchrotron Laboratory
Pasadena, California
PION PRODUCTION IN 300 GEV NUCLEON-NUCLEON COLLISIONS
Keith R. Watson* 
and
Denis Hankins*
June 5, 1961
*University of Southern California
Supported in part by the U.S. Atomic Energy Commission 
Contract No. Α Τ ( 11 - 1 )-68
1
Contents
I Introduction p. 2
II Development p. 4
III Laboratory System Shower Distribution p . 6
IV Effect of Inelasticity on n s p. 10
Appendix p . 12
Abstract
Recent theoretical investigations of high energy nucleon-nucleon
collisions1) support the two-center model of multiple meson production as
discussed by Ciok, et al.2 ) . To facilitate a prediction of the most
probable distribution of shower particles as a  function of laboratory
energy, Farley's kinematical treatment of the "two-fireball" model is
employed 3). A  statistical determination of the multiplicity of pion 
(including neutrals) production per collision is assumed to be valid with 
possible corrections suggested through restrictions imposed by experi­
mental values of the inelasticity, Κπ .
Acknowledgments
The authors would like to take this opportunity to thank 
Professors Matthew Sands and Harold Ticho for making several interesting 
comments and for participating in many useful discussions.
2
I Introduction
A n  analysis of multiple pion production induced by nucleon-nucleon 
collisions is developed on the basis of a two fireball statistical model. 
At 300 Gev primary energy, the statistical model based purely upon phase 
space arguments indicates that about six pions per fireball are created 
per collision. Experimental evidence however, shows that the mean pion 
energy is essentially almost independent of excitation energy (i.e., 0.5 
Gev) in the excited nucleon system. This coupled with an average number 
of six pions (multiplicity of 12), predicts an inelasticity about twice 
as large as that observed experimentally. This suggests, therefore, that 
the number of pions produced per collision (on a statistical basis) should 
be modified b y  matrix element considerations.
Experimental cosmic ray data appears to suggest that the value for 
the inelasticity is in the neighborhood of 0.3 to 0.4 at 300 Gev lab
energy4 ) , at the same time supporting the two fireball concept. In order 
for this value to agree with this model, it is necessary to reduce either 
ns (multiplicity) or the average pion energy in the excited nucleon sys-
tem Ē . Some experiments favor values of Ē as low as 0.4, instead of the
usually accepted value ~ 0 . 5 4). However, this reduction in Ē is still 
not sufficient to lower Kπ to the "correct" value. It is apparent th e r e ­
-fore that ns must be lowered in order to obtain better agreement. Pure 
phase space and statistical equilibrium arguments (a la Fermi) yields a 
value ~ 12 pions/collision (i.e., 6 pions per fireball), hence matrix 
element, or non-equilibrium considerations must be employed in order to 
lower this value.
3
More recent w o r k  based upon "The Eightfold Way"5) has shown that 
the elastic scattering cross section approaches a constant value of 
approximately 40 millibarns at infinite energy1 ) . In addition, experi­
-me n t a l  cosmic ray data also suggests that the inelastic cross section
approaches a constant value in the vicinity of 20 millibarns4). This
indicates, therefore, that the probability per collision of creating
"vector mesons" is about one-third. If we make the further assumption
that single vector meson production is more probable than multiple vector
meson production* , we should expect approximately one pion to be produced
per collision. (This arises from decays such as ρ  →  2π, ω  → 3π.) By
making the further plausible assumption that the average pion energy in
the excited nucleon system is almost constant ~ 0.5 Gev, we may arrive at
an asymptotic value of the inelasticity 0.2, which is in good agreement
with the high energy experimental results6 ) ·
A  direct consequence of lowering the value of n s is to lower the 
excitation energy of the fireballs. This automatically increases the 
value of ϒ , and hence pushes the narrow cone pion distribution to higher 
energies. This is apparently undesirable since some experiments indicate 
that the pion energy spectrum cuts off at energies lower than that pr e ­
-dicted by the Farley model with ns ~ 12. At the present time, there
appears to be no wa y  of resolving this paradox* * .
* This is supported by  recent experiments at 4 Gev; however, this energy 
is probably not large enough to guarantee this statement.
*One possibility is currently being investigated by H. Ticho.
4
In the development, Farley's model wi l l  be exploited in an attempt 
to obtain approximate energy and angular distributions of pions in the 
laboratory.
II Development
We shall consider an inelastic collision of an incoming primary 
nucleon (described b y  energy E p  and Lorentz factor ϒ p ) wi t h  a nucleon 
stationary in the laboratory system (LS). Let βc denote the velocity of 
the center of mass system (CS) where
(1)The two nucleons of rest mass m o and total energy ϒc m o = Ec
approach each other w i t h  equal velocities βc in the CS. After collision,
the mass of each nucleon is increased to ϒ-1 Ec = m o* the velocities of
the two excited nucleons being reduced to β. The energy relation thus 
becomes:
(2)Assuming n s /2 pions per nucleon are emitted isotropically with
average energy Ē 3) in the excited nucleon rest system (N S ) , the excita­
-tion energy m a y  be written in the form
(3)
FIG. 1 KINEMATICS AFTER COLLISION
6
If we let ϒ f and ϒ b describe the motion in the LS of the excited 
nucleons which are emitted in the same and the opposite direction to βp 
in the CS (Fig. 1 ) , it can be easily shown that
(4)
(5)
I LabortySsemhwDiun:(Ep = 300 Gev)
For Ep = 300 Gev, a  Fermi statistical analysis of the meson cloud
yields n s /2 =  6. Thus the E x =  3 curve of Farley's3) energy spectrum of
mesons in the NS ma y  be chosen as that wh i c h  wi l l  yield, under transfor- 
mation, the most probable energy distribution of shower particles in the 
LS. (See Fig. 2.)
The following relations and values will b e  important. From (1 )
(6)
from (3), (2),
(7)
and from (4), (5),
(8)
(9)
Choosing points from Fig. 2 and applying the relation
(10)
RELATIVE PROBABILITY, P (Eπ ) N OF MESON 
EMISSION AS A FUNCTION OF THEIR ENERGIES.
FIG. 2.
8
the following table is obtained:
P(Eπ )N 0.2 0.38 0.42 0.36 0.12 0.02
(Επ )N 0.15 Gev 0.25 0.50 0.565 (mean)
1.0 1.5
(Ρπ )Ν 0.07 Gev 0.21 0.49 0.55 0.99 1.5
TABLE 1
Note that the m e a n  pion energy is in good agreement (in the r e l a ­
-tivistic limit), wi t h  the experimental value of the mean transverse 
momentum PT  = 0.5 Gev/c obtained by Edwards, et al.7), and others.
At 300 Gev, the majority of pions, w h e n  transformed b y  the r e l a ­
tions
(1)and
(12)
will be concentrated in a 1-3 degree cone in the forward direction (see 
A p p e n d i x ) . The energy distribution associated wi t h  the forward cone is 
shown in Fig. 3. Since the backwards cone occupies only a small part 
(0-10 Gev) of the (LS) transformed energy spectrum, Fig. 3 contains 
essentially all the information of interest. The spectrum was transformed 
in the following straightforward manner. Each of the five points in 
Table 1 was transformed according to (1 1 ) for 15° intervals ranging from 
0 to 180 degrees. Since NS emission is isotropic, P(Eπ )N is proportional 
to the solid angle of emission, and therefore weighted as sin θΝ · The
FIG. 3
10
constant θ N curves were added and the resultant curve was renormalized to 
1/2 n s = 6.
IV Effect of Inelasticity on  n s
The inelasticity, Kπ , in a nucleon-nucleon collision, is defined
as that fraction of the total energy available in the CS which is sub-
sequently evaporated in the form of pions. F r o m  Ciok, et al.2 ) , we have
(13)from (3)
hence
and finally
(14)For our particular case n s = 12, Ē ≈  0.56 5, thus Kπ = 0.78.
Gierula, Haskin, and Lohrmann8 ) , and others find values of the
inelasticity which tend towards 0.3 in this energy range. GH L  have used
a low multiplicity, ns =  4, two-center model which closely predicts LS
angular distributions. This work, coupled w i t h  later studies of the 
Bristol experiments by Perkins, seems to demand a re-evaluation of n s ·
A statistical mo d e l  in which only small-angle scattering is impor- 
tant will tend to lower the inelasticity somewhat, since the "core" of
11
the nucleon w i l l  not be excited. Recent w o r k  by Gell-Mann5) has indicated 
that vector mesons may play a strong role in nucleon-nucleon interactions.
On the basis of our extension of this work, it has been shown4) that
(dσ/dΩ)e l a s t i c  is strongly peaked in the forward direction at high energies,
the same behavior is also indicated for the contribution to (dσ/dΩ)inelastic.
This, therefore, suggests that the momentum transfer at high energies is
fairly small, thus reducing the effective excitation energy, and hence
the value of ns and K π derived solely on the basis of the statistical
model. (This again causes an increase in ϒ .)
In conclusion, it should be explained that the two fireball model
of N-N collisions is probably correct for all energies. Sakurai9) has 
suggested that the repulsive nature of the nucleon core should give rise 
to two excited nucleon fireball systems. However, the approximation 
employed in the Fermi statistical model to these separate fireball sys­
-tems is probably invalid since the excitation energy w i l l  tend to remain 
small, even at high energies1 ) . This would probably not be the case for 
N-N systems where a  strong attractive potential exists (possibly even 
large enough to give bound states (e.g., π meson)1 0 ) ), this process would 
clearly enhance the tendency towards a single fireball description, and 
it is hoped that this point can ultimately be checked experimentally.
12
APPENDIX
Number of pions as a function of L S  angle, θ L . For
(1A)Let 1/2 n s be the number of  pions per fireball emitted isotropically 
in the NS. By conservation of particles
(2A)differentiating (1A) and combining wi t h  (2A) gives
o r
(3A)similarly
(4A)Integrai (3)d4
Hence, one may conclude that there are about three pions per col- 
lision in the first degree cone and six in the three degree forward cone, 
on the assumption of six pions per fireball as derived on the basis of 
the statistical model.
13
REFERENCES
1. W. K. R. Watson, B u l . Am. Phys. Soc. II, 6, 372 (1961 ) .
2. P. Ciok, T. Coghen, J. Gierula, R. Holynski, A. Jurak, M. Miesowicz,
T. Saniewska, 0. Stanisz, and J. Pernegr, Nuovo Cimento 8, 166 (1958).
3. F. J. M. Farley, Nuovo Cimento 16, 209 (1 960).
4. D. H. Perkins, Elementary Particle and Cosmic Ra y  Physics, V, (1 960), 
No. Holland Pub. Co., Amsterdam, pp. 259-360.
5. M. Gell-Mann, (to be published in Phys. Rev.) 1961 .
6. G. Bertolino, Nuovo Cimento 2, 1130 (1955).
7. B. Edwards, J. Losty, K. Pinkau, D. H. Perkins, and J. Reynolds, 
Phil. Mag. 3, 237 (1958).
8. J. Gierula, D. M. Haskin, and E. Lohrmann, Phys. Rev. 122, 626 (1 961 ) .
9. J. J. Sakurai, Ann. Phys. (Sept. 1960).
10. P. K a u s , et a l . , B u l . Am. Phys. Soc. II, 6, 377 (1961 ).




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Pion Production in 300 Gev Nucleon-Nucleon Collisions

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