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The formation zones and the large pT production on nuclei
Volume 67B, number 4
PHYSICS LETTERS
THE FORMATION
ZONES AND THE LARGE PT PRODUCTION
25 April 1977
ON NUCLEI
G. WILK Instttute for Nuclear Research, Hoka 69, Warsaw,Poland
Recewed 19 October 1977 The idea of formatmn zone as mtroduced to the hadromc multiproductlon by Stodolsky has been apphed to the descrlptmn of the large PT production on nuclei. It was found that the formation zone concept has to be supplemented by addltmnal assumptions to be in agreement with present experimental data. The notion of the formation zone (FZ) was introduced a long time ago m quantum electrodynamics [1 ]. Recently it has been applied to the hadronic multiproduction processes [2]. In [2] a suggestion was made that the so called "abnormal" nuclear production at large PT as observed by Cronin's group [3] can be a confirmation of the existence of FZ in hadroproductlon. In this note we intend to check this possibility In a more quantitative way. We refer to [1,2] for the derivations and applications of the FZ concept; for our purposes it is enough to specify it in the following way: FZ defined for a given particle h 2 produced by a particle h 1 is a distance ~"measured from the point of interaction along the direction of motion of the h 2 after which the h 2 can be observed as an independent physical object (well defined particle). As the notion of the point of interaction is reasonable only in quantum electrodynamics (from where FZ originated), when applied to hadronlc processes we understand FZ has probably no meaning when 7- > c; c being a parameter responsible for our lack of knowledge of what precisely is going on in the hadromc interactions inside the V ~ c 3 volume (c ~ 1 fm). For the particle h 2 with mass/l, momentum p = (PT, PL), energy co and rapidity y produced by the particle h 1 with rapi&ty Y and velocity 3 (all m the LAB frame) the FZ is given, in analogy with quantum electrodynamlcs, by the formula: P r = r ~ , p T) = ~ ( ~ , - ; ~ P L )
1 |// /aT [ , t g h 2 y + ,T=X/~+p2;
p2 cosh Y • /~2 cosh2y c o s h ( r - y )
3 = tgh Y .
" '
(1)
The meaning o f T ( y , PT) is more clear when it is shown on the phase space diagram for h 1 + N ~ h 2 + anything reaction (see fig. I). All particles coming m above the curve for given r can be observed only at distances z > r. Those with (PT, Y) below the given r are stall forming one state - we will call it the leading hadronic state - LHS. The "abnormal" nuclear production at large PT [3] is a phenomenon of copious production o f the particles with pT ~ 2+3 GeV/c m inclusive reactions on nuclei: h 1 + A ~ h 2 + anything. If we parametrlse the nuclear inclusive distribution I ( A ) = Edo A/d3p in terms of I ( A = 1) and parameter a = a ( p T , . ). I(A) = A~I(A = 1),
(2)
then (see fig. 2) a > 1 for pT>~ 2 + 3 GeV/c. It is contrary to the normal expectation: a ~ 1, a = 1 only if the nucleons all acted independently and there were no shielding effects. There are several explanations of this phenomenon in literature. The multiscatterlng effects [ 4 - 6 ] , the idea of collective "tube" excitations [7, 8], the "violent collision" approach [9] and the possible enhanced behaviour of the nuclear wave function [10] are the explanations used. All of them are bale to explain present data up to PT ~ 3 + 4 GeV/c, for larger PT where a is levelling off (or even turning down) only the models [ 4 - 6 ] can probably give reasonable predictions. In [2] it was suggested that, because the FZ shrinks as the PT of emitted particle increases, it is natural to image the high PT production as incoherent process taking part in the whole volume of nucleus, so a-~ 1. Then a > 1 could be possibly attributed to cascading. To check this idea in a more quantitative way we have to supplement it by some model of propagation of the 443
Volume 67B, number 4
PHYSICS LETTERS
25 April 1977
r, ffm f2
12
1t 8
10 6
0~ 08
2 01 -t
0
l
Z
$
q
S
6.
~.
#.
Fig. 1. The boundary of the phase space for p + N ~ 7r+ anything at ELA B = 300 GeV. Y = 6.4 xs the rapidity of the mcoming proton. The curves of r(y, PT) = const are also shown. The dashed curve corresponds to the fixed angle of emlssmn of secondaries m [3] ; 0LA B = 0.077 rad.
kmpinging particle through the nucleus [11 ]. The simplest possibility is a LHS formed in the first elastic interaction with a given mean free path X2 propagating through the nucleus and emitting, between the successive collisions, particles h 2 for which the r ( h 2 ) •( a i (a I - the mtercollision distances). After the last colhsion a = o% hence all the particles are emitted as in the production on single nucleon. We can easily derwe an expression for R ( A , y , PT) = I(A)/I(1) in terms o f the mean number of collisions ~ and the mean intercollislon distance ~:
t
~
~
q
s
~(6ev/c)
Fig. 2. The results for a (PT, A) as gwen by formula (6). The curves 1-3 correspond go A = 185,48 and 9 respectwely and # = 0.14 GeV. The dashed curve Is for A = 184 and/~ = 1 0 GeV. In all cases ELA B = 300 GeV. There Is no substantial energy dependence when coming to ELA B = 200 GeV and ELA B = 400 GeV. For other parameters see the text. All the data points iie inside the shaded area. V ~ A2/3- (A1/3 - Xl) rather than V ~ A 1 left for incoherentl reactions or ~ = -(a~nel/a~nel. 1,~ -1~ )A2/3" (A /3 _ ;kl) rather than ~ = o ] a n ~ / a ~ A. Since )~1 = 2 + 3 fm (compared with 2R ~ 5 + 1~ fm), the effect is quite substantial. One should then expect a ( p T ) = a (PT, A) =
E1 ln(1 - X 1 A - 1 / 3 ) ] + ~ -j
(5) < 1,
PT>PTa for all finite A. More detailed calculation would give US:
1 for r > : R(A,y, pT)= l + f f - 1)0 [ ~ - r ( y , pT)] = {~ for r < ~ " (3) (We neglected the cascading o f secondaries and energy taken away at each step b y the particles emitted.) On the first sight the result is really encouraging: for given Y a n d P T~ andR(A,y,pT ) = 1, for pT ) PTa all particles h 2 have r < ~, so R ( A , y , p T ) = ~. Then we have: ~ a inel~
a(pT>PTa) = In [ ~hlA / + /rrmel ] ~hl
lnR(A,y,pT) in A
= 1
'
(4)
N-
if only ~ = A(om~;/oLnel ) [12]. But, whenever one ItlJ.~ It lZ~X has propagation throug,h the nucleus, the first interaction (the one producing LHS) takes part (in the mean) tot from the front edge o f at the distance )~1 = 1/P°htN the nucleus. (The nucleus is taked as a uniform sphere of density p and radius R = 1.23 A 1/3.) As m our picture only LHS emits secondaries it is then volume 444
R
Vmax (b)
2~r0f bdb v=2 ~ R(A'y'PT)= 1 + --C--
Pv
v--1
X ~ f(E,y, PT)O [a i - r ( y , P T ) ] , i=1
(6)
(C - normahzation constant). Three ingredients needed are clearly exposed: (i) The FZ comes via the theta function. (ii) The LHS-nucleon Interactions are not necessarily the same as the h 1 -nucleon ones; the (unknown) function f(E, y, PT) takes it into account. (fii) The assumed model o f propagation o f h 1 and LHS through the nucleus is specified by Pv =Pv((al)) • The simplest possibility is h 1 propagating with )h, then after the first interaction the LHS with X2 (~kl,2 = 1/po~°t2N) is formed. We have also assumed no secondary
Volume 67B, number 4
PHYSICS LETTERS
interactions in the distance z = c from the given interaction point. Because o f that there is a hmited number of collisions Umax(b ) for any given impact parameter b. As one can see in fig. 2 our fears were quite justified. Using p = 0.13 fro, o htotN = 32 rob, OLHSN tot = 25mb (or X1 = 2.4 fm, X2= 3.1 fm~ together with the simplest possible choice f(E, y, PT) = 1 we received curves 1 - 3 for different A. This A dependence o f a ( p T ) is rather weak, especially when compared with data error bars, but still indicates that, at least in our approach, o~(PT) as given by (2) is not the best A-rodependent parametrisatmn of I ( A ) . Other parameters were c = 1 fm and the mass of the partMe h2/a = 0.14 GeV. For all these "reasonable" values of parameters our predictions are too low. Even the possible uncertainty o f In romel/omel t hlA/ hlNJ1 = 0.69 + 0.74 [13] is of no help - o~(pT, A) will be still below 1. As we have in our disposal two unknown species' f(E,y, pT ) and the model of propagation o f the incoming particle and produced LHS It is rather easy to get c~= 1 (or even to fit the data). It has not been shown here but it would correspond to quite "unreasonable" values of c, f, OLHS, p (¢ = 2 + 4 f m , f = 8 + 15 depending on PT and/1, OLHS and p such that )k2 = (1/OOLHS) 0.5 fm). They can be possibly explained only in the models mentioned previously (e, f b y [ 7 - 9 ] ; OLHS, O by [10]). There are, however, two characteristic features m our predictions, persisting through all possible changes of parameters mentioned above and, because of that, coming from the FZ concept. It is the plateau in a(py, A) for large PT and possible dip structure for PT ~ 1.0 GeV/e. (This last appears whenever tl = 0 5 + 1.5 GeV). The first is (at least partially) observed, the second should be subject to the experimental test. We conclude, that these two features o f the data, if experimentally confirmed, could be most easily and naturally explained b y the F Z concept which, however, has to be supplemented b y some models
25 April 1977
o f f ( E , y, PT) and Pv to fit experimental points for large PT (and other nuclear data not mentioned here). As for cascading o f secondaries we checked that the full cascade (but without LHS, only with FZ imposed on each particle produced) gives R (A) = ((n)A/(n)N) 1.2. All increase in R (A) is coming from the y < Ymax/2 region only, whereas in [3] y ~ Ymax/2, so we expect the effect of cascade to be rather small in our case.
I wish to express my gratitude to Prof. S. Pokorska for valuable discussions and for reading the manuscript.
References [1] L. Landau and I. Pomeranchuk, Dokl Akad. Nauk SSR 92 (1953) 535,735. Up to data review of the subject ISpresented by. M.I. Ryazanov, Usp. FIZ. Nauk 114 (1974) 393 (Enghsh translation. Soy. Phys. Usp. 17 (1975) 815. [2] L. Stodolsky, Max-Planck-Instltut preprmt, MPI-PAE (PTh23) 75 (Munchen, 1975). [3] J.W. Cronm et al., Phys. Rev D l l (1975) 3105. [4] J. Pumphn and E. Yen, Phys. Rev. D13 (1976) 1812 [5] P.M Flshbane, J. Kotsoms and J.S. Trefil, Umversity of Vtrgmia prepnnt (t976). [6] H.J Ktihn, Phys. Rev. D13 (1976) 2948. [7] S. Fredrlksson, Royal Institute of Technology preprmt, Stockholm, TRITA-TFY-75-13 [8] Y. Afek, G. Berland, G. Eflam and A. Dar, Technoin preprmt, PH-76-12. [9] Meng Ta-chung, Berhn preprint ITR-SB-76-7 [10] A Krzywlckl, Orsay prepnnt LPTPE 76(1), to be pubhshed In Phys Rev. D. [11 ] L. Bertocchl, m. Proc. of the Vlth Int. Conf on High energy physics and nuclear structure, Sata Fe and Los Alamos, 1975. [12] K. Gottfned, m. Proc. of the Vth Int. Conf. on High energy physics and nuclear structure, Uppsala, 1974. [13] W. Busza, in. Proc. of the VIth Int. Conf. on High energy physics and nuclear structure, Santa Fe and Los Alamos, 1975.
445
PHYSICS LETTERS
THE FORMATION
ZONES AND THE LARGE PT PRODUCTION
25 April 1977
ON NUCLEI
G. WILK Instttute for Nuclear Research, Hoka 69, Warsaw,Poland
Recewed 19 October 1977 The idea of formatmn zone as mtroduced to the hadromc multiproductlon by Stodolsky has been apphed to the descrlptmn of the large PT production on nuclei. It was found that the formation zone concept has to be supplemented by addltmnal assumptions to be in agreement with present experimental data. The notion of the formation zone (FZ) was introduced a long time ago m quantum electrodynamics [1 ]. Recently it has been applied to the hadronic multiproduction processes [2]. In [2] a suggestion was made that the so called "abnormal" nuclear production at large PT as observed by Cronin's group [3] can be a confirmation of the existence of FZ in hadroproductlon. In this note we intend to check this possibility In a more quantitative way. We refer to [1,2] for the derivations and applications of the FZ concept; for our purposes it is enough to specify it in the following way: FZ defined for a given particle h 2 produced by a particle h 1 is a distance ~"measured from the point of interaction along the direction of motion of the h 2 after which the h 2 can be observed as an independent physical object (well defined particle). As the notion of the point of interaction is reasonable only in quantum electrodynamics (from where FZ originated), when applied to hadronlc processes we understand FZ has probably no meaning when 7- > c; c being a parameter responsible for our lack of knowledge of what precisely is going on in the hadromc interactions inside the V ~ c 3 volume (c ~ 1 fm). For the particle h 2 with mass/l, momentum p = (PT, PL), energy co and rapidity y produced by the particle h 1 with rapi&ty Y and velocity 3 (all m the LAB frame) the FZ is given, in analogy with quantum electrodynamlcs, by the formula: P r = r ~ , p T) = ~ ( ~ , - ; ~ P L )
1 |// /aT [ , t g h 2 y + ,T=X/~+p2;
p2 cosh Y • /~2 cosh2y c o s h ( r - y )
3 = tgh Y .
" '
(1)
The meaning o f T ( y , PT) is more clear when it is shown on the phase space diagram for h 1 + N ~ h 2 + anything reaction (see fig. I). All particles coming m above the curve for given r can be observed only at distances z > r. Those with (PT, Y) below the given r are stall forming one state - we will call it the leading hadronic state - LHS. The "abnormal" nuclear production at large PT [3] is a phenomenon of copious production o f the particles with pT ~ 2+3 GeV/c m inclusive reactions on nuclei: h 1 + A ~ h 2 + anything. If we parametrlse the nuclear inclusive distribution I ( A ) = Edo A/d3p in terms of I ( A = 1) and parameter a = a ( p T , . ). I(A) = A~I(A = 1),
(2)
then (see fig. 2) a > 1 for pT>~ 2 + 3 GeV/c. It is contrary to the normal expectation: a ~ 1, a = 1 only if the nucleons all acted independently and there were no shielding effects. There are several explanations of this phenomenon in literature. The multiscatterlng effects [ 4 - 6 ] , the idea of collective "tube" excitations [7, 8], the "violent collision" approach [9] and the possible enhanced behaviour of the nuclear wave function [10] are the explanations used. All of them are bale to explain present data up to PT ~ 3 + 4 GeV/c, for larger PT where a is levelling off (or even turning down) only the models [ 4 - 6 ] can probably give reasonable predictions. In [2] it was suggested that, because the FZ shrinks as the PT of emitted particle increases, it is natural to image the high PT production as incoherent process taking part in the whole volume of nucleus, so a-~ 1. Then a > 1 could be possibly attributed to cascading. To check this idea in a more quantitative way we have to supplement it by some model of propagation of the 443
Volume 67B, number 4
PHYSICS LETTERS
25 April 1977
r, ffm f2
12
1t 8
10 6
0~ 08
2 01 -t
0
l
Z
$
q
S
6.
~.
#.
Fig. 1. The boundary of the phase space for p + N ~ 7r+ anything at ELA B = 300 GeV. Y = 6.4 xs the rapidity of the mcoming proton. The curves of r(y, PT) = const are also shown. The dashed curve corresponds to the fixed angle of emlssmn of secondaries m [3] ; 0LA B = 0.077 rad.
kmpinging particle through the nucleus [11 ]. The simplest possibility is a LHS formed in the first elastic interaction with a given mean free path X2 propagating through the nucleus and emitting, between the successive collisions, particles h 2 for which the r ( h 2 ) •( a i (a I - the mtercollision distances). After the last colhsion a = o% hence all the particles are emitted as in the production on single nucleon. We can easily derwe an expression for R ( A , y , PT) = I(A)/I(1) in terms o f the mean number of collisions ~ and the mean intercollislon distance ~:
t
~
~
q
s
~(6ev/c)
Fig. 2. The results for a (PT, A) as gwen by formula (6). The curves 1-3 correspond go A = 185,48 and 9 respectwely and # = 0.14 GeV. The dashed curve Is for A = 184 and/~ = 1 0 GeV. In all cases ELA B = 300 GeV. There Is no substantial energy dependence when coming to ELA B = 200 GeV and ELA B = 400 GeV. For other parameters see the text. All the data points iie inside the shaded area. V ~ A2/3- (A1/3 - Xl) rather than V ~ A 1 left for incoherentl reactions or ~ = -(a~nel/a~nel. 1,~ -1~ )A2/3" (A /3 _ ;kl) rather than ~ = o ] a n ~ / a ~ A. Since )~1 = 2 + 3 fm (compared with 2R ~ 5 + 1~ fm), the effect is quite substantial. One should then expect a ( p T ) = a (PT, A) =
E1 ln(1 - X 1 A - 1 / 3 ) ] + ~ -j
(5) < 1,
PT>PTa for all finite A. More detailed calculation would give US:
1 for r > : R(A,y, pT)= l + f f - 1)0 [ ~ - r ( y , pT)] = {~ for r < ~ " (3) (We neglected the cascading o f secondaries and energy taken away at each step b y the particles emitted.) On the first sight the result is really encouraging: for given Y a n d P T
a(pT>PTa) = In [ ~hlA / + /rrmel ] ~hl
lnR(A,y,pT) in A
= 1
'
(4)
N-
if only ~ = A(om~;/oLnel ) [12]. But, whenever one ItlJ.~ It lZ~X has propagation throug,h the nucleus, the first interaction (the one producing LHS) takes part (in the mean) tot from the front edge o f at the distance )~1 = 1/P°htN the nucleus. (The nucleus is taked as a uniform sphere of density p and radius R = 1.23 A 1/3.) As m our picture only LHS emits secondaries it is then volume 444
R
Vmax (b)
2~r0f bdb v=2 ~ R(A'y'PT)= 1 + --C--
Pv
v--1
X ~ f(E,y, PT)O [a i - r ( y , P T ) ] , i=1
(6)
(C - normahzation constant). Three ingredients needed are clearly exposed: (i) The FZ comes via the theta function. (ii) The LHS-nucleon Interactions are not necessarily the same as the h 1 -nucleon ones; the (unknown) function f(E, y, PT) takes it into account. (fii) The assumed model o f propagation o f h 1 and LHS through the nucleus is specified by Pv =Pv((al)) • The simplest possibility is h 1 propagating with )h, then after the first interaction the LHS with X2 (~kl,2 = 1/po~°t2N) is formed. We have also assumed no secondary
Volume 67B, number 4
PHYSICS LETTERS
interactions in the distance z = c from the given interaction point. Because o f that there is a hmited number of collisions Umax(b ) for any given impact parameter b. As one can see in fig. 2 our fears were quite justified. Using p = 0.13 fro, o htotN = 32 rob, OLHSN tot = 25mb (or X1 = 2.4 fm, X2= 3.1 fm~ together with the simplest possible choice f(E, y, PT) = 1 we received curves 1 - 3 for different A. This A dependence o f a ( p T ) is rather weak, especially when compared with data error bars, but still indicates that, at least in our approach, o~(PT) as given by (2) is not the best A-rodependent parametrisatmn of I ( A ) . Other parameters were c = 1 fm and the mass of the partMe h2/a = 0.14 GeV. For all these "reasonable" values of parameters our predictions are too low. Even the possible uncertainty o f In romel/omel t hlA/ hlNJ1 = 0.69 + 0.74 [13] is of no help - o~(pT, A) will be still below 1. As we have in our disposal two unknown species' f(E,y, pT ) and the model of propagation o f the incoming particle and produced LHS It is rather easy to get c~= 1 (or even to fit the data). It has not been shown here but it would correspond to quite "unreasonable" values of c, f, OLHS, p (¢ = 2 + 4 f m , f = 8 + 15 depending on PT and/1, OLHS and p such that )k2 = (1/OOLHS) 0.5 fm). They can be possibly explained only in the models mentioned previously (e, f b y [ 7 - 9 ] ; OLHS, O by [10]). There are, however, two characteristic features m our predictions, persisting through all possible changes of parameters mentioned above and, because of that, coming from the FZ concept. It is the plateau in a(py, A) for large PT and possible dip structure for PT ~ 1.0 GeV/e. (This last appears whenever tl = 0 5 + 1.5 GeV). The first is (at least partially) observed, the second should be subject to the experimental test. We conclude, that these two features o f the data, if experimentally confirmed, could be most easily and naturally explained b y the F Z concept which, however, has to be supplemented b y some models
25 April 1977
o f f ( E , y, PT) and Pv to fit experimental points for large PT (and other nuclear data not mentioned here). As for cascading o f secondaries we checked that the full cascade (but without LHS, only with FZ imposed on each particle produced) gives R (A) = ((n)A/(n)N) 1.2. All increase in R (A) is coming from the y < Ymax/2 region only, whereas in [3] y ~ Ymax/2, so we expect the effect of cascade to be rather small in our case.
I wish to express my gratitude to Prof. S. Pokorska for valuable discussions and for reading the manuscript.
References [1] L. Landau and I. Pomeranchuk, Dokl Akad. Nauk SSR 92 (1953) 535,735. Up to data review of the subject ISpresented by. M.I. Ryazanov, Usp. FIZ. Nauk 114 (1974) 393 (Enghsh translation. Soy. Phys. Usp. 17 (1975) 815. [2] L. Stodolsky, Max-Planck-Instltut preprmt, MPI-PAE (PTh23) 75 (Munchen, 1975). [3] J.W. Cronm et al., Phys. Rev D l l (1975) 3105. [4] J. Pumphn and E. Yen, Phys. Rev. D13 (1976) 1812 [5] P.M Flshbane, J. Kotsoms and J.S. Trefil, Umversity of Vtrgmia prepnnt (t976). [6] H.J Ktihn, Phys. Rev. D13 (1976) 2948. [7] S. Fredrlksson, Royal Institute of Technology preprmt, Stockholm, TRITA-TFY-75-13 [8] Y. Afek, G. Berland, G. Eflam and A. Dar, Technoin preprmt, PH-76-12. [9] Meng Ta-chung, Berhn preprint ITR-SB-76-7 [10] A Krzywlckl, Orsay prepnnt LPTPE 76(1), to be pubhshed In Phys Rev. D. [11 ] L. Bertocchl, m. Proc. of the Vlth Int. Conf on High energy physics and nuclear structure, Sata Fe and Los Alamos, 1975. [12] K. Gottfned, m. Proc. of the Vth Int. Conf. on High energy physics and nuclear structure, Uppsala, 1974. [13] W. Busza, in. Proc. of the VIth Int. Conf. on High energy physics and nuclear structure, Santa Fe and Los Alamos, 1975.
445