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Inclusive hadron-nucleus scattering at high energy
Volume 87B, number 4
PHYSICS LETTERS
19 November 1979
INCLUSIVE HADRON-NUCLEUS SCATTERING AT HIGH ENERGY ~ T. FUJITA and J. HUFNER Institut fgir Theoretische Physik, Universitdt Heidelberg, Fed. Rep. Germany and Max.Planck-Institut fgir Kernphysik, Heidelberg, Fed. Rep. Germany Received 14 August 1979
The multiple scattering series of Glauber and Matthiae for hadron-nucleus scattering is summed to a simple analytical expression. It reproduces quantitatively the main features of the experimental cross section for the reaction 4He + A Z 4He + X at 1 GeV/nucleon.
What physics can be learned from inclusive hadron -nucleus scattering in the energy region of several GeV? To be specific, we put this question concerning a reaction for which experimental data exist. But derivation and results are more general. The cross section for the reaction 4 H e + A z -> 4He
+ X,
102
I
(1)
has been measured at the SATURNE [1] and at the BEVALAC [2] for different targets AZ and for several energies of the incident a-particle. The data exhibit some striking features: The cross section do/d2q where q is the transversemomentum of the outgoing a, is dominated by an exponential decay do/d2q = Bexp (-q/qo). This regularity is immediately recognized in figs. 1 and 2. The data for 4He as a target show some additional modulation of the exponential decay. On fig. 2, the exponential decay is visible over five orders of magnitude, the deviations at low momenta (q < 0.2 GeV/c) arise from the elastic scattering contributions to (1). The decay constant q0 of the exponential takes the value q0 = 90 MeV/c and does neither depend on the momentum of the incident a (between 4 and 12 GeV/c) nor on the target nucleus. Also the scale factor B of the exponential decay does not change with energy and increases by only a factor three from 12C to 208pb as targets. To the best of our knowledge, there have never been similar data before which extend to Supported in part by a grant from the German federal ministry for Research and Technology (BMFT).
such a large momentum transfer. Therefore the regularities described above have not previously emerged so clearly. They aroused our curiosity. The use of 4He as
1
~+OC
= o~+X
101 $
'~o
4,32 GeWc
"d
¢.D
lO°
~.o~10-1
;\
10-2
10-3
0
I
0,5
-
I
1,0
q [GeV/c]
Fig. 1. The inclusive inelastic cross section for a - a scattering for an incident m o m e n t u m of 4.32 GeV/c. The differential cross section is given as a function of the transverse m o m e n t u m q o f the outgoing ~,. The points are taken from ref. [1], the solid curve is the result o f the calculation. The n u m b e r s 1 to 4 in circles indicate the locations where single, double, etc. a N scattering dominates. (The n u m b e r s correspond to the location o f the saddle p o ~ t ns, eq. (12).)
327
Volume 87B, number 4
PHYSICS LETTERS
~,
104
I
I
I
~
19 November 1979
10 6
I
I
I
105
10 3 L--" o~ + C -.,,-o~ + X 102
~ + Pb --,,- ~ + X 10z.
7 GeV/c
101
103
c~ 100
..~10 2
7 GeV/e
e~
E
E
~ i0-I
01
-o 10_2
~ 100
10-3 I:-
(~ ~\
10.4 L-
0
~,V
0.5 1.0 q [GeVlc]
-]
10-1
-J
lo -2
1.5
%
103 0
0.5 1.0 q [GeVlc]
1.5
Fig. 2. The data and the results of the calculation for a - C and ct-Pb inclusive scattering (here the inclusive scattering includes the elastic part which dominates at low q). The experimental points of ref. [2] are converted to da/d2q as explained in the text (eq. (14)). The solid line for 12C corresponds to the assumption of a gaussian for the target. The dashed line for 12C and the solid line for Pb are computed for a density with a linearized surface. The meaning of the numbers 1 to 6 is the same as in fig. 1. the incident hadron makes the reaction particularly clean: 4He has no particle stable excited state. Hence there is no significant contribution to the cross section eq. (1) in which the 4He is excited and decays on its way between the target and the detector. Elastic a - n u c l e u s scattering at high energies seems well understood [3] and we need not treat it. The formalism for inclusive h a & o n - n u c l e u s scattering has been worked out in the classical paper by Glauber and Matthiae [4]. We remain within this formalism. We show: The multiple scattering series of ref. [4] can be summed to give an analytical expression. It reproduces quantitatively the main features of the experiment and permits insight into the physics of reaction eq. (1). Glauber and Matthiae represent the differential inclusive inelastic cross section by the series A
do ( a + A z _ ~ a + X)=n~__1 onFn(q).
d2q
=
(2)
the ground state. The index n in the sum denotes the number o f a-nucleon ( a - N ) collisions which occur dur. ing a-nucleus scattering. The cross section for n a - N collisions is called a n and F n (q) represents the momentum distribution (in transverse direction) after such a collision: =
] fd2bto
328
exp(--o,,totN
T(b)),
(3)
where OaN el and "aN .tot are the elastic and total a - N cross sections, respectively. The nuclear thickness function T(b) is defined as T(b) =
fdzp(b,z), fd2br(b)=A,
(4)
where p(r) is the density distribution o f the target nucleus folded with the matter distribution o f 4He. The function Fn(q) is obtained-by folding n times the differential elastic a - N cross sections daeal/d2q n
The final state X of the target can be any state except
r(b)],
Fn(q) = Cn
el
fi~=i d2qi ~d°c~N d2qi
6(2)
(q - ~ qi ). (5) i=1
Volume 87B, number 4
PHYSICS LETTERS 10 3
The normalization constant Cn is chosen so that fd2qFn(q) = I. In the derivation of eqs. (2) to (5) the independent particle model is used to describe the target nucleus and nothing has to be assumed about the structure of the projectile. It is "elementary". The sum eq. (2) can be evaluated numerically without problems. Our aim, however, is to obtain an analytical expression for the sum eq. (2) in order to exhibit the salient features of the physics. According to experiment (fig. 3 and ref. [5] ), the elastic a - N cross section can be approximated by a gaussian e x p ( - q 2 / p2) with P0 = 185 MeV/c which describes well the low q behaviour, where the cross section drops over three orders of magnitude. For gaussians the folding eq. (5) is simple and gives
1
(
Fn(q)=lmp2 exp 1
19 November 1979 i
c¢ + p - - . . o ~ . X
7 G eV/c \ \
# 10 1
(gaussian),
(8a)
db 2/du
(uniform),
(8b)
(linear surface).
(8c)
db2/du =-(XtR)2/3u -1/3
The following notation is used: @2) is the mean square radius of the gaussian density. If the uniform distribution is characterized by a constant density PO, the mean free path X of an a-particle in this nucleus is X= 1 = P0 ° t ~ (typically X ~ 0.5 fm). Formula (8c) is derived for a nucleus where the shape of the nuclear surface is approximated by a linear fall-off with r: p(r)o~ O(R - r ) (R - r)/t, where t is the surface thickness (t ~ 2.4 fm practically independent of A). Eq. (8c) is valid for X '< t. Expressions (8a)-(8c) can be summarized into the form
db2 /du = - L 2u t~ ,
- aoo. d
lO o
\
"bo
10-1 t
I
I
0.5
1.0
q [GeWe]
Fig. 3. The shape of the a - N cross section from ref. [2] and its fit by a gaussian.
Ir L 2
db2/du = _ ~_@2 ) (1/u) - ~-)2u
-
(7)
The integral can be evaluated approximately by converting the integration over db 2 into one over u. Depending on the radial shape of the target density p (r) we find
=
--
2
,t
10-2
un(b) e -u(b) .
q
.,
0
_ ,r 1 .fdb: On n! e n
-
10 2
q2
When we introduce the notation e = ote°l~/oeelN and t~(b) = ~tot " a N T(b), the cross section for n a - N collisions eq. (3) is written as
I
(9)
where the length L is characteristic of the target. Then
f On nr en 0 -
2R/X
L 2 (n+3)! duun+3e -u .~Tr-en n!--(lO)"
The second step is a good approximation as long as n +/~ 2R/X, where 2R/X is the size of the nuclear diameter measured in mean free paths X. Eqs. (10) and (6) are inserted into eq. (2). We calculate first the total inelastic inclusive cross section of reaction (1). If e >> 1, the first term in the multiple scattering series dominates and the integration yields
Otot(a + A z ~ a + X)= rrL2 OaNiOa , tot .
(11)
This is the first result of our paper. To obtain the differential cross section, especially for large q, the terms with n > 1 are important. The sum eq. (2) is evaluated by converting it into an integral over n and evaluating it by the saddle point method. The saddle point n s depends on q and is obtained from the equation
(~/an) (nln e + (l/n)(q/po)2)ln=ns = 0 .
(12)
For a given q of the final a, ns(q) represents the most probable number of a - N collisions, the a has undergone. They are indicated on fig. 1 and 2 in the circles. Then elementary algebra yields 329
Volume 87B, number 4
PHYSICS LETTERS
sample the integrated inclusive cross section for fixed q. Therefore we assume the relation
(do/d2q) (a +A z ~ a + X) = (L2/p2)exp[-2(ln e)l/2q/po]
(13)
× (n/In e) 1/2 (n s +/3) !/n s !X~s. This equation is the second result of our paper. Its accuracy is checked by numerically evaluating eq. (2) and is found excellent except for n s = 1 (q ~< p0 x/Tn e). We also checked numerically the approximations leading to eq. (2) and found them very good. The inclusive a-nucleus cross section eq. (13) scales by L 2/p2, where the length L is characteristic of the target nucleus and P0 describes the a - N cross section. The q dependence is dominated by the exponential which contains no information on the target. F o r p 0 = 190 MeV/c and e = 3 [6], the decay constant is 90 MeV/c in good agreement with experiment. The shape of the target nucleus influences the exponent/3, eqs. (8) and (9), and introduces an additional weak dependence o n q via ns(q). The absolute magnitude, given by L 2 depends weakly on the target. If we use eq. (8c), do/d2q =A 2/9 and predicts a factor two between C and Pb (observed is a factor three or four). Figs. 1 and 2 show a quantitative comparisonbetween the data and the analytical formula eq. (13). Without any free parameter, the magnitude and the overall slope for the a - - a cross section is reproduced (fig. 1). We cannot explain the additional diffraction type structure. In fig. 2 the shape of the cross sections and the relative magnitude between C and Pb are completely determined and fairly well reproduced. But there is a problem with the absolute magnitude. The experimental paper ref. [2] presents Edo/d2qdp as a function of q for fixed total momentum p equal to the incident momentum. Since the total momentum loss of the a in the inclusive reaction is of the order of a few 100 MeV/c according to ref. [1 ] the acceptance in momentum p of ref. [2] seems sufficiently large to
330
19 November 1979
E d3 a/d2 q dp = (E/ Ap) do/d2 q ,
(14)
where the size Ap reflects the experimental acceptance We obtain the value E/Ap = 10 by comparing the a - p cross sections from ref. [2] where the left-hand side of eq. (14) is measured, with ref. [5] which measures do/d2q. The value of E/Ap from a = N is then used to convert the experimental a-nucleus cross sections on C and Pb to do/d2q. In this way also the absolute magnitudes of the cross sections in fig. 2 are understood without a free parameter. We summarize: The main features of the inclusive a-nucleus cross sections can be understood qualitatively and quantitatively without new assumptions. The essential information necessary to describe the data relates to the a - N cross section, and very little need to be known about the target. The weak target dependence may follow from the inclusive nature of reaction: Since we do not observe the final state of the target, we cannot expect to learn much about the target. We thank J. Duflo, L. Goldzahl, J. Knoll, H. Pirner and F. Houin for several discussions. E. Morawcsik typed several versions of the manuscript and E. Seitz drew the figures.
References [1] J. Duflo et al., contribution to the eight Intern. Conf. on High-energy physics and nuclear structure (Vancouver, 1979) and private communication. [2] L.M. Anderson, Ph.D. Thesis (1977), Berkeley report LBL-6769. [3] G. Fgldt and I. Hulthage, Nucl. Phys. A316 (1979) 253. [4] R.J. Glauber and G. Matthiae, Nucl. Phys. B21 (1970) 135. [5] H. Palevsky et al., Phys. Rev. Lett. 18 (1967) 1200. [6] J.P. Meyer, Astron. Astrophys. Suppl. 7 (1972) 417.
PHYSICS LETTERS
19 November 1979
INCLUSIVE HADRON-NUCLEUS SCATTERING AT HIGH ENERGY ~ T. FUJITA and J. HUFNER Institut fgir Theoretische Physik, Universitdt Heidelberg, Fed. Rep. Germany and Max.Planck-Institut fgir Kernphysik, Heidelberg, Fed. Rep. Germany Received 14 August 1979
The multiple scattering series of Glauber and Matthiae for hadron-nucleus scattering is summed to a simple analytical expression. It reproduces quantitatively the main features of the experimental cross section for the reaction 4He + A Z 4He + X at 1 GeV/nucleon.
What physics can be learned from inclusive hadron -nucleus scattering in the energy region of several GeV? To be specific, we put this question concerning a reaction for which experimental data exist. But derivation and results are more general. The cross section for the reaction 4 H e + A z -> 4He
+ X,
102
I
(1)
has been measured at the SATURNE [1] and at the BEVALAC [2] for different targets AZ and for several energies of the incident a-particle. The data exhibit some striking features: The cross section do/d2q where q is the transversemomentum of the outgoing a, is dominated by an exponential decay do/d2q = Bexp (-q/qo). This regularity is immediately recognized in figs. 1 and 2. The data for 4He as a target show some additional modulation of the exponential decay. On fig. 2, the exponential decay is visible over five orders of magnitude, the deviations at low momenta (q < 0.2 GeV/c) arise from the elastic scattering contributions to (1). The decay constant q0 of the exponential takes the value q0 = 90 MeV/c and does neither depend on the momentum of the incident a (between 4 and 12 GeV/c) nor on the target nucleus. Also the scale factor B of the exponential decay does not change with energy and increases by only a factor three from 12C to 208pb as targets. To the best of our knowledge, there have never been similar data before which extend to Supported in part by a grant from the German federal ministry for Research and Technology (BMFT).
such a large momentum transfer. Therefore the regularities described above have not previously emerged so clearly. They aroused our curiosity. The use of 4He as
1
~+OC
= o~+X
101 $
'~o
4,32 GeWc
"d
¢.D
lO°
~.o~10-1
;\
10-2
10-3
0
I
0,5
-
I
1,0
q [GeV/c]
Fig. 1. The inclusive inelastic cross section for a - a scattering for an incident m o m e n t u m of 4.32 GeV/c. The differential cross section is given as a function of the transverse m o m e n t u m q o f the outgoing ~,. The points are taken from ref. [1], the solid curve is the result o f the calculation. The n u m b e r s 1 to 4 in circles indicate the locations where single, double, etc. a N scattering dominates. (The n u m b e r s correspond to the location o f the saddle p o ~ t ns, eq. (12).)
327
Volume 87B, number 4
PHYSICS LETTERS
~,
104
I
I
I
~
19 November 1979
10 6
I
I
I
105
10 3 L--" o~ + C -.,,-o~ + X 102
~ + Pb --,,- ~ + X 10z.
7 GeV/c
101
103
c~ 100
..~10 2
7 GeV/e
e~
E
E
~ i0-I
01
-o 10_2
~ 100
10-3 I:-
(~ ~\
10.4 L-
0
~,V
0.5 1.0 q [GeVlc]
-]
10-1
-J
lo -2
1.5
%
103 0
0.5 1.0 q [GeVlc]
1.5
Fig. 2. The data and the results of the calculation for a - C and ct-Pb inclusive scattering (here the inclusive scattering includes the elastic part which dominates at low q). The experimental points of ref. [2] are converted to da/d2q as explained in the text (eq. (14)). The solid line for 12C corresponds to the assumption of a gaussian for the target. The dashed line for 12C and the solid line for Pb are computed for a density with a linearized surface. The meaning of the numbers 1 to 6 is the same as in fig. 1. the incident hadron makes the reaction particularly clean: 4He has no particle stable excited state. Hence there is no significant contribution to the cross section eq. (1) in which the 4He is excited and decays on its way between the target and the detector. Elastic a - n u c l e u s scattering at high energies seems well understood [3] and we need not treat it. The formalism for inclusive h a & o n - n u c l e u s scattering has been worked out in the classical paper by Glauber and Matthiae [4]. We remain within this formalism. We show: The multiple scattering series of ref. [4] can be summed to give an analytical expression. It reproduces quantitatively the main features of the experiment and permits insight into the physics of reaction eq. (1). Glauber and Matthiae represent the differential inclusive inelastic cross section by the series A
do ( a + A z _ ~ a + X)=n~__1 onFn(q).
d2q
=
(2)
the ground state. The index n in the sum denotes the number o f a-nucleon ( a - N ) collisions which occur dur. ing a-nucleus scattering. The cross section for n a - N collisions is called a n and F n (q) represents the momentum distribution (in transverse direction) after such a collision: =
] fd2bto
328
exp(--o,,totN
T(b)),
(3)
where OaN el and "aN .tot are the elastic and total a - N cross sections, respectively. The nuclear thickness function T(b) is defined as T(b) =
fdzp(b,z), fd2br(b)=A,
(4)
where p(r) is the density distribution o f the target nucleus folded with the matter distribution o f 4He. The function Fn(q) is obtained-by folding n times the differential elastic a - N cross sections daeal/d2q n
The final state X of the target can be any state except
r(b)],
Fn(q) = Cn
el
fi~=i d2qi ~d°c~N d2qi
6(2)
(q - ~ qi ). (5) i=1
Volume 87B, number 4
PHYSICS LETTERS 10 3
The normalization constant Cn is chosen so that fd2qFn(q) = I. In the derivation of eqs. (2) to (5) the independent particle model is used to describe the target nucleus and nothing has to be assumed about the structure of the projectile. It is "elementary". The sum eq. (2) can be evaluated numerically without problems. Our aim, however, is to obtain an analytical expression for the sum eq. (2) in order to exhibit the salient features of the physics. According to experiment (fig. 3 and ref. [5] ), the elastic a - N cross section can be approximated by a gaussian e x p ( - q 2 / p2) with P0 = 185 MeV/c which describes well the low q behaviour, where the cross section drops over three orders of magnitude. For gaussians the folding eq. (5) is simple and gives
1
(
Fn(q)=lmp2 exp 1
19 November 1979 i
c¢ + p - - . . o ~ . X
7 G eV/c \ \
# 10 1
(gaussian),
(8a)
db 2/du
(uniform),
(8b)
(linear surface).
(8c)
db2/du =-(XtR)2/3u -1/3
The following notation is used: @2) is the mean square radius of the gaussian density. If the uniform distribution is characterized by a constant density PO, the mean free path X of an a-particle in this nucleus is X= 1 = P0 ° t ~ (typically X ~ 0.5 fm). Formula (8c) is derived for a nucleus where the shape of the nuclear surface is approximated by a linear fall-off with r: p(r)o~ O(R - r ) (R - r)/t, where t is the surface thickness (t ~ 2.4 fm practically independent of A). Eq. (8c) is valid for X '< t. Expressions (8a)-(8c) can be summarized into the form
db2 /du = - L 2u t~ ,
- aoo. d
lO o
\
"bo
10-1 t
I
I
0.5
1.0
q [GeWe]
Fig. 3. The shape of the a - N cross section from ref. [2] and its fit by a gaussian.
Ir L 2
db2/du = _ ~_@2 ) (1/u) - ~-)2u
-
(7)
The integral can be evaluated approximately by converting the integration over db 2 into one over u. Depending on the radial shape of the target density p (r) we find
=
--
2
,t
10-2
un(b) e -u(b) .
q
.,
0
_ ,r 1 .fdb: On n! e n
-
10 2
q2
When we introduce the notation e = ote°l~/oeelN and t~(b) = ~tot " a N T(b), the cross section for n a - N collisions eq. (3) is written as
I
(9)
where the length L is characteristic of the target. Then
f On nr en 0 -
2R/X
L 2 (n+3)! duun+3e -u .~Tr-en n!--(lO)"
The second step is a good approximation as long as n +/~ 2R/X, where 2R/X is the size of the nuclear diameter measured in mean free paths X. Eqs. (10) and (6) are inserted into eq. (2). We calculate first the total inelastic inclusive cross section of reaction (1). If e >> 1, the first term in the multiple scattering series dominates and the integration yields
Otot(a + A z ~ a + X)= rrL2 OaNiOa , tot .
(11)
This is the first result of our paper. To obtain the differential cross section, especially for large q, the terms with n > 1 are important. The sum eq. (2) is evaluated by converting it into an integral over n and evaluating it by the saddle point method. The saddle point n s depends on q and is obtained from the equation
(~/an) (nln e + (l/n)(q/po)2)ln=ns = 0 .
(12)
For a given q of the final a, ns(q) represents the most probable number of a - N collisions, the a has undergone. They are indicated on fig. 1 and 2 in the circles. Then elementary algebra yields 329
Volume 87B, number 4
PHYSICS LETTERS
sample the integrated inclusive cross section for fixed q. Therefore we assume the relation
(do/d2q) (a +A z ~ a + X) = (L2/p2)exp[-2(ln e)l/2q/po]
(13)
× (n/In e) 1/2 (n s +/3) !/n s !X~s. This equation is the second result of our paper. Its accuracy is checked by numerically evaluating eq. (2) and is found excellent except for n s = 1 (q ~< p0 x/Tn e). We also checked numerically the approximations leading to eq. (2) and found them very good. The inclusive a-nucleus cross section eq. (13) scales by L 2/p2, where the length L is characteristic of the target nucleus and P0 describes the a - N cross section. The q dependence is dominated by the exponential which contains no information on the target. F o r p 0 = 190 MeV/c and e = 3 [6], the decay constant is 90 MeV/c in good agreement with experiment. The shape of the target nucleus influences the exponent/3, eqs. (8) and (9), and introduces an additional weak dependence o n q via ns(q). The absolute magnitude, given by L 2 depends weakly on the target. If we use eq. (8c), do/d2q =A 2/9 and predicts a factor two between C and Pb (observed is a factor three or four). Figs. 1 and 2 show a quantitative comparisonbetween the data and the analytical formula eq. (13). Without any free parameter, the magnitude and the overall slope for the a - - a cross section is reproduced (fig. 1). We cannot explain the additional diffraction type structure. In fig. 2 the shape of the cross sections and the relative magnitude between C and Pb are completely determined and fairly well reproduced. But there is a problem with the absolute magnitude. The experimental paper ref. [2] presents Edo/d2qdp as a function of q for fixed total momentum p equal to the incident momentum. Since the total momentum loss of the a in the inclusive reaction is of the order of a few 100 MeV/c according to ref. [1 ] the acceptance in momentum p of ref. [2] seems sufficiently large to
330
19 November 1979
E d3 a/d2 q dp = (E/ Ap) do/d2 q ,
(14)
where the size Ap reflects the experimental acceptance We obtain the value E/Ap = 10 by comparing the a - p cross sections from ref. [2] where the left-hand side of eq. (14) is measured, with ref. [5] which measures do/d2q. The value of E/Ap from a = N is then used to convert the experimental a-nucleus cross sections on C and Pb to do/d2q. In this way also the absolute magnitudes of the cross sections in fig. 2 are understood without a free parameter. We summarize: The main features of the inclusive a-nucleus cross sections can be understood qualitatively and quantitatively without new assumptions. The essential information necessary to describe the data relates to the a - N cross section, and very little need to be known about the target. The weak target dependence may follow from the inclusive nature of reaction: Since we do not observe the final state of the target, we cannot expect to learn much about the target. We thank J. Duflo, L. Goldzahl, J. Knoll, H. Pirner and F. Houin for several discussions. E. Morawcsik typed several versions of the manuscript and E. Seitz drew the figures.
References [1] J. Duflo et al., contribution to the eight Intern. Conf. on High-energy physics and nuclear structure (Vancouver, 1979) and private communication. [2] L.M. Anderson, Ph.D. Thesis (1977), Berkeley report LBL-6769. [3] G. Fgldt and I. Hulthage, Nucl. Phys. A316 (1979) 253. [4] R.J. Glauber and G. Matthiae, Nucl. Phys. B21 (1970) 135. [5] H. Palevsky et al., Phys. Rev. Lett. 18 (1967) 1200. [6] J.P. Meyer, Astron. Astrophys. Suppl. 7 (1972) 417.