- Email: [email protected]
Cumulative production of particles in high energy collisions
Volume 93B, number 1,2
PHYSICS LETTERS
2 June 1980
CUMULATIVE PRODUCTION OF PARTICLES IN HIGH ENERGY COLLISIONS ~ G. BERLAD, A. DAR and G. EILAM Department of Physics, Technion-Israel Institute of Technology, Haifa, Israel Received 27 March 1980
We show that the patton recombination model of hadron production combined with a quark-parton model of the nucleus describe cumulative meson production in high energy collisions with nuclei remarkably well.
The fragmentation of nuclei in high energy collisions into elementary particles with momenta far exceeding the average momentum per nucleon in the nucleus is one of the most interesting phenomena in high energy physics. This phenomenon, so called cumulative production, was first observed by Baldin and co-workers [1]. Since then it has been extensively studied in high energy particle-nucleus [2] and nucleus-nucleus collisions [3] at accelerator energies, and in cosmic rays collisions up to energies of several TeV [4], but so far no satisfactory quantitative explanation of the phenomenon has been demonstrated [5]. Recently parton recombination and fragmentation models have been successfully applied to the fragmentation of hadrons into low PT-large x particles in high energy particle collisions [6]. In this letter we show that when these models are combined with a quarkparton model of a nucleus in a high momentum frame [7] they describe well cumulative particle production. The patton recombination/fragmentation models assume that a projectile (p) fragments when one of its constituent quarks collides in the target (T). The spectator quarks that escape collisions in the target retain their original fraction x of the c.m. momentum and hadronize (or fragment and then hadronize) by recombining with slow quarks (x ~ 0) from the sea. If the produced hadron shares a valence quark in common with the projectile then at low PT the dominant proWork supported in part by the United States-Israel Binational Science Foundation. 86
duction mechanism is quark recombination and the x-distribution of the fragment will be that of the quark in the projectile, i.e. [8] o T (p -+ h) - E
d3o dp3
(pT -+ h) = Ct~ (OpT -- OqT) qp (X)
(1) where Chq at a fixed PT is a constant that depends only on q and h. OpT and OqT are the total absorption cross sections off T for P and q respectively. OpT OqT is the cross section that q escapes collisions with T while at least one of the other constituent quarks of p does collide with T (otherwise p does not fragment), qp(X) is the longitudinal momentum distribution of q in p as observed for instance in deep inelastic lepton scattering [9]. Eq. (1) describes well nucleon (1'4) fragmentation into n-*, K ± , K s with low PT and large x in high energy nucleon-nucleon [10], nucleon-meson [11] and nucleon-nucleus [8] collisions. To generalize eq. (1) to the fragmentation of high energy nuclei we assume that only those nucleons that collide can fragment into mesons and baryons (other than nucleons), while the spectator of the nucleus fragments only into nucleons and nuclei. Eq. (1) can then be written as -
-
a T (A -~ h) =- E
d3o dP 3
(AT --~h) ~ CI~(ONT -- OqT)qn(x ) (2)
where x is the fraction of the c.m. momentum of A carried by q with distribution qA(x) normalized such that flqA(X)dx is the total number of q-quarks in A.
Volume 93B, number 1,2
PHYSICS LETTERS
qA can be extracted from deep inelastic lepton scattering on nuclei, but since no such measurements at large values of x are presently available we shall estimate it from a quark parton model o f a nucleus [7]. According to current belief Quantum Chromo Dynamics (QCD) is the underlying field theory o f strong interactions. In QCD coloured quarks interact strongly via the exchange o f coloured gluons. These interactions are believed to bind and confine coloured quarks into colour singlets. Being colourless the nucleons can not interact strongly via an exchange o f a coloured gluon. That is why nuclear forces are strong within the nucleons but relatively weak at distances larger than the nucleon radius. When a nucleus is viewed in a high m o m e n t u m frame it is Lorentz contracted. All the nucleons in an imaginary tube t with a cross section o of a nucleon, drawn along the momentum axis, are contracted into a narrow disk of the same cross section. Due to the longitudinal shortening all the nucleons in the tube will now communicate easily; their communication in the transverse direction will not be affected. Lorentz contraction will simultaneously increase the longitudinal m o m e n t u m components of the nucleons due to the uncertainty principle and reduce their interaction strength due to asymptotic freedom. Thus nucleons in a contracted tube will behave like a free Fermi gas, and the total momentum of the tube will be distributed among them according to phase space rules [7] Nt(x ) =
i(2i - 1)(2i - 2)x(1 - x) 2i- 3 .
(3)
Here i is the number o f nucleons in the tube and Nt(x ) is the probability that a nucleon carries a fraction x of its longitudinal momentum. It satisfies the sum rules f ~ N t ( x ) d x = i and f$xUt(x)dx = 1. qt(x) is then given by the convolution 1
qt(x)
=f 0
1
f qN(Xl)Xt(x2) 6(x - XlX2) dXldX 2 0
1
= f qN(x/x')Nt(x')dx'/x',
(4)
x
where qN and qt are the x-distributions of q in N and t respectively, qN can be extracted from deep inelastic lepton scattering. E.g. u N and d N can be represented by
2 June 1980
x)3/X/x, = Un(X ) = (315/256)(1 - x)4/x/Cx.
Up(X) = d n ( x ) = ( 3 5 / 1 6 ) ( 1 dp(x)
(5)
By substituting eqs. (5) into (4) one obtains for a tube containing (Z/A)i protons and (1 - Z/A)i neutrons: ut(x ) =
(Z/A )ui(x) + (1 - Z/A )d i (x) , (6)
d t ( x ) = (1 -
Z/A)ui(x ) + (Z/A)di(x),
where
ui(x)_ 35 3 ( 1 - x ) 2i+t 16 (2i + 1)x/x
(7)
X F ( 2 . 5 , 2 i - 2, 2i + 2; 1 - x ) , 315
dg(x) = ~
12(1 - x ) 2i+2 (2i + 1)(2i + 2)x,~
(s)
X F ( 3 . 5 , 2 i - 2, 2i + 3; 1 - x ) .
F(a, b, c; z) is the confluent hypergeometric function. For x --* 1, ui(x ) ~ (1 - x ) 2i+1. This result can also be obtained from the dimensional counting rules [12] by counting as elementary spectators to a quark in a tube the other two valence quarks in the same nucleon and the other i - 1 nucleons in the tube. Note that the structure functions (7), (8) replace the universality assumption or the formerly used structure functions in the Tube Model [13]. The abundance of tubes o f cross section o that contain exactly i nucleons in a nucleus of A nucleons can be estimated [13] using an independent particle model of the nucleus:
T(b) = f ~ p(b, z)dz is the nuclear thickness at impact parameter b and p is the nuclear density function normalized such that fpd3r = I. (Through this paper we will use o = Opp = 33 mb and a standard WoodsSaxon nuclear density function p ~ {1 + exp [(r - R)/ d ] ) -1 w i t h R = 1.1A 1/3 fm and d = 0.54 fm). qA(x) can now be written as A
qA(x) = . ~
t=l
N(i,A)qt(xi)
(10)
87
Volume 93B, number 1,2
PHYSICS LETTERS
10 z
2 June 1980
10s pD ~-n-(180") ; 8.6 GeV/c
101
--
•
77" ÷
0
7/'"
V
CTM
SR.
- -
CTM
CAL.
10o _
%
> %
(.9
>~ i0-I _ -ol'm W •,o i ' o LIJ
I0":' -
i0-~ -
o.1
\ 10'
I
I o.1
I 0.2
I
I
I
I 0.3 0.4 T. [GeV]
I
I i I 0.6 0.5
I
I o.7
o2
o3 0.4 m,, [ GeV]
0.5
0.6
0.7
Fig. 2. Comparison between experimental results [ 1 ] on pA rr+(180°), A = D, C, Cu and Pb at Plab = 8.4 GeV/c and eq. (12).
10 2
p i l e 4 - - - yr ( 1 8 0 " )
; PLAS=8.6
and the cross section (2) for e.g. A ~ 7r+ can then be
GeV/c
w r i t t e n as 101
•
/7"+
o
Tr-
A
OT(A
no N >
E
10":
0.1
0.2
0.3 %r[GeV]
88
~N(i, A)ut(xi), (11)
i=1
where x i is the fraction of the c.m. momentum of tube i carried by 7r+. Eqs. (2) and (11) have been derived under the assumption that quarks are not attenuated within the incident nucleus. Such an attenuation can be incorporated in eqs. (2) and (11) using the additive quark model [7]. For instance for T = p, eq. (12) should be
10 0
10"
r t + ) = C~u (O'NT -- OqT )
04
05
0.6
0.7
Fig. 1. a. Comparison between experimental results [14] on pD --* rr-+(180 °) at Plab = 8.6 GeV/c, eq. (14) (full line), and the sum rule (15) (SR). In figs. 1 - 3 Trr =Err - mrr, and the arrow indicates the maximum kinetic energy o f a n in a collision with a stationary nucleon, b. Comparison between experimental results [14] on p4He -~ rr-+(180 °) at Plab = 8.6 and eq. (12).
GeV/c
Volume 93B, number 1,2
PHYSICS LETTERS
rewritten as
10 4 ,
A op(A ~ 7r+) = 2C~UOqp i=~1?(i)N(i,A)
ut(xi) ,
(12)
where Oqp = (1/2)O~p = (1/3)Opp and ~,(i) = 9 [6 + 2/9) l - 7(2/3y]/14i. op(A -+ rr) can be obtained from eq. (12) by replacing u t by dt. For medium and heavy weight nuclei and not too large x-values eq. (12) can be approximated by Op(A ~ 7r+) ~
2 June 1980
2CUoqpT(v)(A/v)ut(Xv),
(13)
where v = A Opp/OpA is the average nuclear thickness and A/v is the effective number of nuclear tubes. Let us first apply eq. (12) to deuteron fragmentation. Using a Hulth6n wave function to estimate N(i, D) eq. (12) can be written as
~ o
Edd'~" (pToIIIl~"lr (160")) PL:B~ 400 GeV/c
~
- - S U M OVER TUBES AVERAGE TUBE P=3.96
103
o
~
-
-
~T-
~¢j 10z >
E \~
_
.~-~ 101
op(D ~ rr-+) = 0.79 {Op(p -~ rr+) + ap(p -~ rr-))
10 0
(14) + 0.21Op(t ~ 7r-+) . where we assume that Op(p ~ 7r-+) = Op(n ~ 7r~) and where t stands for a tube of 2 nucleons. For A1 nuclei v = 2 and eqs. (13) and (14) yield the approximate sum rule op(D ~ rr+-) = 0.79{Op(p -~ 7r+) + Op(p ~ rr-)} +
10-1
I
0.2
I '~ ]
0.4
I
I
I
I
0o6 0.8 T~r[GeV]
I\~1
1.0
\
I
1.2
I
1.4
10
(15)
R_= o'(pA~ rr+)/A (:r(pLi%"'n'÷)/6
0.016Op(Al ~ 7r+-) ,
which does not depend on a specific model of tube fragmentation. In fig. la we compare experimental data [14] on pD -~ 7r-+(180°) at Plab = 8.6 GeV/c and predictions (14) and (15). The R.H.S. of (15) was evaluated from experimental data on pp ~ lr+-(180 °) at Plab = 8.6 GeV/c [14] and on pAl -~ zr~(180 °) at Plab = 8.4 GeV/c [1]. The R.H.S. of (14) was evaluated with Op(p -~ 7r-+) taken from experiment and with gp(t -+ n"+-)= CUoqp')'(2)[u2(x2) + d2(x2) ] . C u = C d was determined from pp -+ 7r-+(180°) at Plab 8.6 GeV/c [14]. In fig. lb we compare prediction (12) and experimental data on p4He ~ 7r-+(180°) at Plab = 8.6 GeV/c. In fig. 2 we compare prediction (12) and ex-
I
PLAB=400 GeV/c P.,, =0.68 GeV/c
90*
135°
tr
=
Fig. 3. a. Comparison between experimental results [ 15 ] on pTa ~ ~r-+(160 °) at Plab = 400 GeV/c, eq. (12) (full line), and approximate predictions as given by eq. (13) (dotted line). b. Comparison between experimental results on the ratio PA/P6Li where PA = apo~Ed 3 .°o/d3 a/dp 3 ~pA ._), lr) for Pn =0.68GeVand0n=90 , 1 3 5 and 160 a t P l a b = 3 0 0 GeV/c, and our predictions based on eq. (12).
160 °
1
t =~
I
10
A
100
I
I
I I II
1000
89
Volume 93B, number 1,2
PHYSICS LETTERS
perimental data [1] on pA ~ rr-*(180°) A = D, C, Cu and Pb, at Plab = 8.4 GeV/c. In fig. 3a we compare predictions (12), (13) and experimental data [15] on pTa ~ zr*-(160°) at Plab = 400 GeV/c. In our calculations we used [161 C ff ~ Cff = (0.4 ± 0.1) [exp(10pT 2) + 0.45 e x p ( - 2 . 7 p T 2 ) ] [GeV2/c 3 ]-1 which together with eqs. (1) and (5) describe well the world data on zr-* production at low PT (PT ~ 1 GeV/c) and large x in high energy (s > 100 GeV 2) pp collisions. In fig. 3b we compare the A-dependence predicted by eq. (12) and experimental results [15] on 7r-* production from nuclei at P~r = 0.68 GeV/c and 07r = 90 °, 135 ° and 160 ° by 400 GeV/c protons. Figs. 1 - 3 demonstrate that our model correctly predicts the x, A and 0-dependence of cumulative backward production of mesons from nuclear targets by high energy protons. Our model can be further improved by better parametrizations of Up and dp especially at low energies where scaling is broken, by including the contribution from recombination of wounded quarks (quarks that suffer inelastic collisions), by introducing x-dependent recombination probabilities etc. The model can be easily generalized to other projectiles and fragments such as p, K -+, K0, A, D, 3He, etc. These improvements and generalizations will be discussed elsewhere.
References [1] A.M. Baldin et al., Soy. Jour. Nucl. Phys. 18 (1974) 41; 20 (1975) 629.
90
2 June 1980
[2] See for instance, A.M. Baldin, Particles and Nuclei 8, 429 (1977); A.M. Baldin, Proc. 19th Int. Conf. on High energy physics, Tokyo 1978, eds. S. Homma et al. (Phys. Soc. Japan Tokyo 1979) p. 455, and references therein. [3] J. Papp et al., Phys. Rev. Lett. 34 (1975) 601. [4] M.O. Azaryan et al., Soy. Jour. Nucl. Phys. 26 (1977) 72. [5] Fermi motion models, even with unconventionally long momentum tails cannot explain the data. See for instance R.H. Landau and M. Gyulassy, Phys. Rev. C19 (1979) 149 and references therein. Significant progress was made however by I.A. Schmidt and R. Blankenbecler, Phys. Rev. 15D (1977) 3321 and by F. Takagi, Phys. Rev. D19 (1979) 2612. See however L.S. Schroeder et al., preprint LBL-9434 (1979). [6] See for instance, L. Van Hove, CERN Prepfint Th. 2628 (1979). [7] G. Berlad, A. Dar and G. Eilam, Technion preprint PH79-71 (1979). [8] See for instance G. Berlad and A. Dar, Technion preprint PH-79-69 (1979). [9] W. Ochs, Nucl. Phys. Bl18 (1977) 397. We neglect QCD scaling violation. [10] G. Berlad, S. Dado and A. Dar, to be published. [11] J. Hanlon et al., Stony Brook preprint (1979). [12] R. Blankenbecler and S.J. Brodsky, Phys. Rev. D10 (1974) 2973; J. Gunion, Phys. Rev. D10 (1974) 242. [13] G. Berlad, A. Dar and G. Eilam, Phys. Rev. D13 (1976) 161; Y. Afek et al., Phys. Rev. D20 (1979) 1160. [14] A.M. Baldin et al., JINR Report E1-11368 Dubna (1978), [15] Yu.D. Bayukov et al., University of Pennsylvania report UPR-0058E (November 1978). [16] W.H. Sims et al., Nucl. Phys. B41 (1972) 317.
PHYSICS LETTERS
2 June 1980
CUMULATIVE PRODUCTION OF PARTICLES IN HIGH ENERGY COLLISIONS ~ G. BERLAD, A. DAR and G. EILAM Department of Physics, Technion-Israel Institute of Technology, Haifa, Israel Received 27 March 1980
We show that the patton recombination model of hadron production combined with a quark-parton model of the nucleus describe cumulative meson production in high energy collisions with nuclei remarkably well.
The fragmentation of nuclei in high energy collisions into elementary particles with momenta far exceeding the average momentum per nucleon in the nucleus is one of the most interesting phenomena in high energy physics. This phenomenon, so called cumulative production, was first observed by Baldin and co-workers [1]. Since then it has been extensively studied in high energy particle-nucleus [2] and nucleus-nucleus collisions [3] at accelerator energies, and in cosmic rays collisions up to energies of several TeV [4], but so far no satisfactory quantitative explanation of the phenomenon has been demonstrated [5]. Recently parton recombination and fragmentation models have been successfully applied to the fragmentation of hadrons into low PT-large x particles in high energy particle collisions [6]. In this letter we show that when these models are combined with a quarkparton model of a nucleus in a high momentum frame [7] they describe well cumulative particle production. The patton recombination/fragmentation models assume that a projectile (p) fragments when one of its constituent quarks collides in the target (T). The spectator quarks that escape collisions in the target retain their original fraction x of the c.m. momentum and hadronize (or fragment and then hadronize) by recombining with slow quarks (x ~ 0) from the sea. If the produced hadron shares a valence quark in common with the projectile then at low PT the dominant proWork supported in part by the United States-Israel Binational Science Foundation. 86
duction mechanism is quark recombination and the x-distribution of the fragment will be that of the quark in the projectile, i.e. [8] o T (p -+ h) - E
d3o dp3
(pT -+ h) = Ct~ (OpT -- OqT) qp (X)
(1) where Chq at a fixed PT is a constant that depends only on q and h. OpT and OqT are the total absorption cross sections off T for P and q respectively. OpT OqT is the cross section that q escapes collisions with T while at least one of the other constituent quarks of p does collide with T (otherwise p does not fragment), qp(X) is the longitudinal momentum distribution of q in p as observed for instance in deep inelastic lepton scattering [9]. Eq. (1) describes well nucleon (1'4) fragmentation into n-*, K ± , K s with low PT and large x in high energy nucleon-nucleon [10], nucleon-meson [11] and nucleon-nucleus [8] collisions. To generalize eq. (1) to the fragmentation of high energy nuclei we assume that only those nucleons that collide can fragment into mesons and baryons (other than nucleons), while the spectator of the nucleus fragments only into nucleons and nuclei. Eq. (1) can then be written as -
-
a T (A -~ h) =- E
d3o dP 3
(AT --~h) ~ CI~(ONT -- OqT)qn(x ) (2)
where x is the fraction of the c.m. momentum of A carried by q with distribution qA(x) normalized such that flqA(X)dx is the total number of q-quarks in A.
Volume 93B, number 1,2
PHYSICS LETTERS
qA can be extracted from deep inelastic lepton scattering on nuclei, but since no such measurements at large values of x are presently available we shall estimate it from a quark parton model o f a nucleus [7]. According to current belief Quantum Chromo Dynamics (QCD) is the underlying field theory o f strong interactions. In QCD coloured quarks interact strongly via the exchange o f coloured gluons. These interactions are believed to bind and confine coloured quarks into colour singlets. Being colourless the nucleons can not interact strongly via an exchange o f a coloured gluon. That is why nuclear forces are strong within the nucleons but relatively weak at distances larger than the nucleon radius. When a nucleus is viewed in a high m o m e n t u m frame it is Lorentz contracted. All the nucleons in an imaginary tube t with a cross section o of a nucleon, drawn along the momentum axis, are contracted into a narrow disk of the same cross section. Due to the longitudinal shortening all the nucleons in the tube will now communicate easily; their communication in the transverse direction will not be affected. Lorentz contraction will simultaneously increase the longitudinal m o m e n t u m components of the nucleons due to the uncertainty principle and reduce their interaction strength due to asymptotic freedom. Thus nucleons in a contracted tube will behave like a free Fermi gas, and the total momentum of the tube will be distributed among them according to phase space rules [7] Nt(x ) =
i(2i - 1)(2i - 2)x(1 - x) 2i- 3 .
(3)
Here i is the number o f nucleons in the tube and Nt(x ) is the probability that a nucleon carries a fraction x of its longitudinal momentum. It satisfies the sum rules f ~ N t ( x ) d x = i and f$xUt(x)dx = 1. qt(x) is then given by the convolution 1
qt(x)
=f 0
1
f qN(Xl)Xt(x2) 6(x - XlX2) dXldX 2 0
1
= f qN(x/x')Nt(x')dx'/x',
(4)
x
where qN and qt are the x-distributions of q in N and t respectively, qN can be extracted from deep inelastic lepton scattering. E.g. u N and d N can be represented by
2 June 1980
x)3/X/x, = Un(X ) = (315/256)(1 - x)4/x/Cx.
Up(X) = d n ( x ) = ( 3 5 / 1 6 ) ( 1 dp(x)
(5)
By substituting eqs. (5) into (4) one obtains for a tube containing (Z/A)i protons and (1 - Z/A)i neutrons: ut(x ) =
(Z/A )ui(x) + (1 - Z/A )d i (x) , (6)
d t ( x ) = (1 -
Z/A)ui(x ) + (Z/A)di(x),
where
ui(x)_ 35 3 ( 1 - x ) 2i+t 16 (2i + 1)x/x
(7)
X F ( 2 . 5 , 2 i - 2, 2i + 2; 1 - x ) , 315
dg(x) = ~
12(1 - x ) 2i+2 (2i + 1)(2i + 2)x,~
(s)
X F ( 3 . 5 , 2 i - 2, 2i + 3; 1 - x ) .
F(a, b, c; z) is the confluent hypergeometric function. For x --* 1, ui(x ) ~ (1 - x ) 2i+1. This result can also be obtained from the dimensional counting rules [12] by counting as elementary spectators to a quark in a tube the other two valence quarks in the same nucleon and the other i - 1 nucleons in the tube. Note that the structure functions (7), (8) replace the universality assumption or the formerly used structure functions in the Tube Model [13]. The abundance of tubes o f cross section o that contain exactly i nucleons in a nucleus of A nucleons can be estimated [13] using an independent particle model of the nucleus:
T(b) = f ~ p(b, z)dz is the nuclear thickness at impact parameter b and p is the nuclear density function normalized such that fpd3r = I. (Through this paper we will use o = Opp = 33 mb and a standard WoodsSaxon nuclear density function p ~ {1 + exp [(r - R)/ d ] ) -1 w i t h R = 1.1A 1/3 fm and d = 0.54 fm). qA(x) can now be written as A
qA(x) = . ~
t=l
N(i,A)qt(xi)
(10)
87
Volume 93B, number 1,2
PHYSICS LETTERS
10 z
2 June 1980
10s pD ~-n-(180") ; 8.6 GeV/c
101
--
•
77" ÷
0
7/'"
V
CTM
SR.
- -
CTM
CAL.
10o _
%
> %
(.9
>~ i0-I _ -ol'm W •,o i ' o LIJ
I0":' -
i0-~ -
o.1
\ 10'
I
I o.1
I 0.2
I
I
I
I 0.3 0.4 T. [GeV]
I
I i I 0.6 0.5
I
I o.7
o2
o3 0.4 m,, [ GeV]
0.5
0.6
0.7
Fig. 2. Comparison between experimental results [ 1 ] on pA rr+(180°), A = D, C, Cu and Pb at Plab = 8.4 GeV/c and eq. (12).
10 2
p i l e 4 - - - yr ( 1 8 0 " )
; PLAS=8.6
and the cross section (2) for e.g. A ~ 7r+ can then be
GeV/c
w r i t t e n as 101
•
/7"+
o
Tr-
A
OT(A
no N >
E
10":
0.1
0.2
0.3 %r[GeV]
88
~N(i, A)ut(xi), (11)
i=1
where x i is the fraction of the c.m. momentum of tube i carried by 7r+. Eqs. (2) and (11) have been derived under the assumption that quarks are not attenuated within the incident nucleus. Such an attenuation can be incorporated in eqs. (2) and (11) using the additive quark model [7]. For instance for T = p, eq. (12) should be
10 0
10"
r t + ) = C~u (O'NT -- OqT )
04
05
0.6
0.7
Fig. 1. a. Comparison between experimental results [14] on pD --* rr-+(180 °) at Plab = 8.6 GeV/c, eq. (14) (full line), and the sum rule (15) (SR). In figs. 1 - 3 Trr =Err - mrr, and the arrow indicates the maximum kinetic energy o f a n in a collision with a stationary nucleon, b. Comparison between experimental results [14] on p4He -~ rr-+(180 °) at Plab = 8.6 and eq. (12).
GeV/c
Volume 93B, number 1,2
PHYSICS LETTERS
rewritten as
10 4 ,
A op(A ~ 7r+) = 2C~UOqp i=~1?(i)N(i,A)
ut(xi) ,
(12)
where Oqp = (1/2)O~p = (1/3)Opp and ~,(i) = 9 [6 + 2/9) l - 7(2/3y]/14i. op(A -+ rr) can be obtained from eq. (12) by replacing u t by dt. For medium and heavy weight nuclei and not too large x-values eq. (12) can be approximated by Op(A ~ 7r+) ~
2 June 1980
2CUoqpT(v)(A/v)ut(Xv),
(13)
where v = A Opp/OpA is the average nuclear thickness and A/v is the effective number of nuclear tubes. Let us first apply eq. (12) to deuteron fragmentation. Using a Hulth6n wave function to estimate N(i, D) eq. (12) can be written as
~ o
Edd'~" (pToIIIl~"lr (160")) PL:B~ 400 GeV/c
~
- - S U M OVER TUBES AVERAGE TUBE P=3.96
103
o
~
-
-
~T-
~¢j 10z >
E \~
_
.~-~ 101
op(D ~ rr-+) = 0.79 {Op(p -~ rr+) + ap(p -~ rr-))
10 0
(14) + 0.21Op(t ~ 7r-+) . where we assume that Op(p ~ 7r-+) = Op(n ~ 7r~) and where t stands for a tube of 2 nucleons. For A1 nuclei v = 2 and eqs. (13) and (14) yield the approximate sum rule op(D ~ rr+-) = 0.79{Op(p -~ 7r+) + Op(p ~ rr-)} +
10-1
I
0.2
I '~ ]
0.4
I
I
I
I
0o6 0.8 T~r[GeV]
I\~1
1.0
\
I
1.2
I
1.4
10
(15)
R_= o'(pA~ rr+)/A (:r(pLi%"'n'÷)/6
0.016Op(Al ~ 7r+-) ,
which does not depend on a specific model of tube fragmentation. In fig. la we compare experimental data [14] on pD -~ 7r-+(180°) at Plab = 8.6 GeV/c and predictions (14) and (15). The R.H.S. of (15) was evaluated from experimental data on pp ~ lr+-(180 °) at Plab = 8.6 GeV/c [14] and on pAl -~ zr~(180 °) at Plab = 8.4 GeV/c [1]. The R.H.S. of (14) was evaluated with Op(p -~ 7r-+) taken from experiment and with gp(t -+ n"+-)= CUoqp')'(2)[u2(x2) + d2(x2) ] . C u = C d was determined from pp -+ 7r-+(180°) at Plab 8.6 GeV/c [14]. In fig. lb we compare prediction (12) and experimental data on p4He ~ 7r-+(180°) at Plab = 8.6 GeV/c. In fig. 2 we compare prediction (12) and ex-
I
PLAB=400 GeV/c P.,, =0.68 GeV/c
90*
135°
tr
=
Fig. 3. a. Comparison between experimental results [ 15 ] on pTa ~ ~r-+(160 °) at Plab = 400 GeV/c, eq. (12) (full line), and approximate predictions as given by eq. (13) (dotted line). b. Comparison between experimental results on the ratio PA/P6Li where PA = apo~Ed 3 .°o/d3 a/dp 3 ~pA ._), lr) for Pn =0.68GeVand0n=90 , 1 3 5 and 160 a t P l a b = 3 0 0 GeV/c, and our predictions based on eq. (12).
160 °
1
t =~
I
10
A
100
I
I
I I II
1000
89
Volume 93B, number 1,2
PHYSICS LETTERS
perimental data [1] on pA ~ rr-*(180°) A = D, C, Cu and Pb, at Plab = 8.4 GeV/c. In fig. 3a we compare predictions (12), (13) and experimental data [15] on pTa ~ zr*-(160°) at Plab = 400 GeV/c. In our calculations we used [161 C ff ~ Cff = (0.4 ± 0.1) [exp(10pT 2) + 0.45 e x p ( - 2 . 7 p T 2 ) ] [GeV2/c 3 ]-1 which together with eqs. (1) and (5) describe well the world data on zr-* production at low PT (PT ~ 1 GeV/c) and large x in high energy (s > 100 GeV 2) pp collisions. In fig. 3b we compare the A-dependence predicted by eq. (12) and experimental results [15] on 7r-* production from nuclei at P~r = 0.68 GeV/c and 07r = 90 °, 135 ° and 160 ° by 400 GeV/c protons. Figs. 1 - 3 demonstrate that our model correctly predicts the x, A and 0-dependence of cumulative backward production of mesons from nuclear targets by high energy protons. Our model can be further improved by better parametrizations of Up and dp especially at low energies where scaling is broken, by including the contribution from recombination of wounded quarks (quarks that suffer inelastic collisions), by introducing x-dependent recombination probabilities etc. The model can be easily generalized to other projectiles and fragments such as p, K -+, K0, A, D, 3He, etc. These improvements and generalizations will be discussed elsewhere.
References [1] A.M. Baldin et al., Soy. Jour. Nucl. Phys. 18 (1974) 41; 20 (1975) 629.
90
2 June 1980
[2] See for instance, A.M. Baldin, Particles and Nuclei 8, 429 (1977); A.M. Baldin, Proc. 19th Int. Conf. on High energy physics, Tokyo 1978, eds. S. Homma et al. (Phys. Soc. Japan Tokyo 1979) p. 455, and references therein. [3] J. Papp et al., Phys. Rev. Lett. 34 (1975) 601. [4] M.O. Azaryan et al., Soy. Jour. Nucl. Phys. 26 (1977) 72. [5] Fermi motion models, even with unconventionally long momentum tails cannot explain the data. See for instance R.H. Landau and M. Gyulassy, Phys. Rev. C19 (1979) 149 and references therein. Significant progress was made however by I.A. Schmidt and R. Blankenbecler, Phys. Rev. 15D (1977) 3321 and by F. Takagi, Phys. Rev. D19 (1979) 2612. See however L.S. Schroeder et al., preprint LBL-9434 (1979). [6] See for instance, L. Van Hove, CERN Prepfint Th. 2628 (1979). [7] G. Berlad, A. Dar and G. Eilam, Technion preprint PH79-71 (1979). [8] See for instance G. Berlad and A. Dar, Technion preprint PH-79-69 (1979). [9] W. Ochs, Nucl. Phys. Bl18 (1977) 397. We neglect QCD scaling violation. [10] G. Berlad, S. Dado and A. Dar, to be published. [11] J. Hanlon et al., Stony Brook preprint (1979). [12] R. Blankenbecler and S.J. Brodsky, Phys. Rev. D10 (1974) 2973; J. Gunion, Phys. Rev. D10 (1974) 242. [13] G. Berlad, A. Dar and G. Eilam, Phys. Rev. D13 (1976) 161; Y. Afek et al., Phys. Rev. D20 (1979) 1160. [14] A.M. Baldin et al., JINR Report E1-11368 Dubna (1978), [15] Yu.D. Bayukov et al., University of Pennsylvania report UPR-0058E (November 1978). [16] W.H. Sims et al., Nucl. Phys. B41 (1972) 317.