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Overlap function from the multiperipheral model
Volume 50B, number 3
PHYSICS LETTERS
10 June 1974
OVERLAP FUNCTION FROM THE MULTIPERIPHERAL MODEL S. JADACH
Institute of Physics,JagellonianUniversity,Cracow,Poland and J. TURNAU
Institute of NuclearPhysics, Cracow,Poland Received 29 March 1974 Revised manuscript received 12 May 1974 Monte Carlo phase space calculations of the overlap function for Chan-Loskiewicz-Allisonand Chew-Pignotti models were performed up to laboratory momentum 1500 GeV. Slope of the overlap function was found about four times smaller than the experimental value. Shrinkage of the elastic peak is dose to that found in experiment. The qualitative explanation of these results is given, and the discrepancy with the recent estimates obtained by Hamer and Peierls, Hwa, and Henyey is explained. Finally, our analysis shows that the random walk picture of the multiperipheral models is not valid for realistic density of particles in rapidity scale. The measurements of the elastic and inelastic scattering cross-sections at NAL and ISR energies have stimulated a new interest in the problem of the Van Hove overlap function [1]. In particular, several authors discussed recently this problem in the framework of the multiperipheral model [ 2 - 4 ] . In the present letter we report briefly the study of the overlap function in two specific versions of the multiperipheral model, namely the Chan-LoskiewiczAllison and Chew-Pignotti models. The more detailed description of the calculations and of the results will be published elsewhere [13]. Our arguments and c o n clusions are based on exact Monte Carlo calculations performed up to energy 1500 GeV, and taking into account energy and momentum conservation. These calculations show that the slope of the overlap function is much smaller than that of elastic amplitude. Thus they agree with earlier calculations at lower energies, but they are in strong disagreement with conclusions of refs. [ 2 - 4 ] . In order to introduce the reader into the problem let us write down the equation for the imaginary part of the two body (elastic) amplitude, which is direct consequence of the S-matrix unitarity.
lm
A2(paPb;pa,Pb,)
(1)
where A 2 is an elastic scattering amplitude a+b ~ a+b,A n is an amplitude for production n particles a+b -* 1+2+ ... +n, and drn is the Lorentz invariant phase space element. An adequate model of particle production inserted into the right-hand side of eq. (1) has to generate correct amplitude of the elastic scattering, which is believed to be dominantly imaginary in the high energy limit. Thus the eq. (1) provides a test of models for particle production. As it has been shown by Michejda and collaborators [5-8] this test is rather severe and many models do not satisfy it. The RHS of eq. (1) is called the overlap function [1 ]. In the following we consider its inelastic part (n > 2) which determines elastic scattering amplitude up to 10%. We analyzed two types of multiperipheral parametrization for the matrix element. One of them was Chan-Loskiewicz-Allison model [9-10] widely studied at energies below 30 GeV. In this model modulus of the matrix element for n particles production is of the form
t +c°
(2)
where
2 = ~ 2 fdrn An(PaPb; Pl 'P2 "'"Pn)A*(Pa'Pb';Pl "'"Pn)
=(Pi +Pi+1 )2 _(mi +mi +1)2,
ti= (Pa-- k=lPk) 369
Volume 50B, number 3
PHYSICS LETTERS
r
5
10 June 1974
7
"-"
4 3
b
,! 2
3
1
lb
2b
3(3 n
Fig. 1. Partial overlap slopes versus total multiplicity In the
CLA modle at Plab: a-50, b-300, c-1500 GeV. Parameters g, c, a, b were taken fromthe original paper [9]. We have performed Monte Carlo calculations of the inelastic overlap function for the whole total multiplicity spectrum at 50, and 300 GeV, and for multiplicities close to the average at 1500 GeV (estimated from logarithmic fit). We have taken only one multiperipheral graph, and only with meson reggeons exchanged (with intercept 0.5). We disregarded all other graphs with permutated positions of the final particles. The phases of the amplitudes were taken constant. In our calculations we used a new method for generating multiperipheral Monte Carlo events, developed for high energy calculations. Detailed description of the method will be published elsewhere. In figs. l and 2 we present the results of our calculations. As we can see the slopes of partial overlap functions are small in comparison with the slope of the elastic amplitude (6 GeV -2) and decrease with increasing multiplicity of intermediate state, contrary to the prediction of the random walk picture (fig. 1). It is seen also that the obtained shrinkage is comparable to the experimental one (fig. 2). Thus our results agree with calculations at lower energies [7]. They disagree entirely, however, with those of refs. [ 2 - 4 ] , in which it was concluded that (i) the slope of the overlap function in the multiperipheral model is bigger than the experimentally determined, and (ii) the shrinkage is much stronger than experimentally observed. As we shall show below, this difference arises from the fact that authors of refs. [2-4] neglected longitudinal part of the four momentum transfer t L. In the 370
loo
2oo
5oo
looo
2ooo . I'c v ]
ooo
Fig. 2. Slope of the overlap function versus In(s) at average total multiplicity, a-experimental, b-from CLA model, c-from CP model.
multiperipheral model the slope of the overlap function is, roughly speaking, proportional to 1/(ti). In the approximation of the vanishing t L
(ti)~.(tT)= -
~pT k k=l
(3)
Taking into account experimental distribution of the transverse momenta of produced particles we get t ~. - 0 . 1 . From our calculations it appears that for the multiplicities close to average, average longitudinal momentum transfer is around - 1 . 6 GeV 2. Thus formula (3) leads to underestimation of the average t by order of magnitude and, in consequence, to overestimation of the overlap function slope and shrinkage. Let us also point out that the average value of the parameter a i = In (1 +si/bi) which governs the momentum transfer distribution in the CLA model has nothin common with the range of nuclear forces if (t i) > (si). In such evidently not multiperipheral situation a i should be regarded as pure phenomenological parameter. Although bounds on four momentum transfers are very weak, transverse momenta are strongly bounded, and we get their average values consistent with the data. To,see how it happens let us write a formula for longitudinal momentum transfer along ith link in multiperipheral chain
Volume 50B, number 3
PHYSICS LETTERS
2
i
2
(4)
10 June 1974
We have performed the same calculations at the same energies and multiplicities for the simple ChewPignotti model n-1
Taking into account energy momentum conservation and assuming that produced particles are equally spaced in rapidity scale with distance d we can derive the following approximate formula t L ~ (mT) 2 e-d/(l_e-d)
2
(5)
where m T = (m 2 +pT 2)1/2. This approximation is valid for links far from the ends of the multiperipheral chain. The approximate transverse momentum dependence of the amplitude through t L is exp (aeffp T) where ~eff <°~)e-a/(1-e-d) 2. Taking d = In s/
IA n I = i~_l f(si)
exp
(ati),
(6)
In order to reproduce the experimental average transverse momenta we had to take for constant a the value about 0.5-0.4 GeV -2 in rough agreement with CLA value (a i) = 0.4, and in strong disagreement with approximation a = 2/<(pT) 2) = 5.5 GeV -2 used in refs. [2-4]. Comparing the results of the Monte Carlo calculations with CLA and CP models at average total multiplicities for different energies (see fig. 2), we can see that there is not substantial difference between them as far as the overlap function and its shrinkage are concerned. We conclude that flatness of the overlap function cannot be attributed to the particular CLA parametrization but (in the framework of the models with single emission from each vertex) is a general feature of the multiperipheral kinematics. To summarize, we have shown by means of the Monte Carlo calculations for pp collisions at Plab 50, 300, and 1500 GeV that in the Chan-LoskiewiczAllison and Chew-Pignotti models without momentum dependent phases slope of the overlap function is four times smaller than it was found experimentally. Shrinkage was found close to experimental value. These results are in disagreement with calculations based on the approximation in which longitudinal momentum transfers are neglected. We proved that this approximation, which is incidentally the basis of the random walk picture, cannot be justified for multiplicities close to and larger than average. Non negligible longitudinal momentum transfers result from the high particle density in the rapidity scale and energy-momentum conservation. The authors would like to thank Professor A. Bialas for his help and encouragement. They thank also Dr. K. Fialkowski and Professor K. Zalewski for readhag of the manuscript and critical remarks.
371
Volume 50B, number 3
PHYSICS LETTERS
References [1] [2] [3] [4] [5] [6] [7]
372
L. Van Hove, Rev. Mod. Phys. 36 (1964). R.C. Hwa, Phys. Rev. D8 (1973) 1331. C.J. Hamer and R.F. Peierls, Phys. Rev. D8 (1973) 1358. F.S. Henyey, Phys. Lett. B45 (1973) 469. L. Michejda, Nucl. Phys. B4 (1968) 113. L. Michejda, Fortschr. Phys. 16 (1968) 707. L. Michejda, J. Turnau and A. Bialas, Nuovo Cimento A56 (1968) 241.
10 June 1974
[8] A. Jurewicz, L. Michejda, J. Namyslowski and J. Turnau, Nucl. Phys. B29 (1971) 269. [9] H.M. Chan, J. Loskiewicz and W.W.M.Allison, Nuovo Cimento A57 (1968) 93. [10] Z. Ajduk, L. Michejda and W. W6jcik, Acta Phys. Pol. A37 (1970) 285. [11] V.N. Gribov, Yad. Fiz. 9 (1969) 640; J.D. Bjorken, preprint SLAC-PUB-905 (1971). [12] G.F. Chew and A. Pignotti, Phys. Rev. 176 (1968) 2112. [13] S. Jadach and J. Turnau, preprint TPJU-3/74, February 1974; Acta Phys. Pol. B, to be published.
PHYSICS LETTERS
10 June 1974
OVERLAP FUNCTION FROM THE MULTIPERIPHERAL MODEL S. JADACH
Institute of Physics,JagellonianUniversity,Cracow,Poland and J. TURNAU
Institute of NuclearPhysics, Cracow,Poland Received 29 March 1974 Revised manuscript received 12 May 1974 Monte Carlo phase space calculations of the overlap function for Chan-Loskiewicz-Allisonand Chew-Pignotti models were performed up to laboratory momentum 1500 GeV. Slope of the overlap function was found about four times smaller than the experimental value. Shrinkage of the elastic peak is dose to that found in experiment. The qualitative explanation of these results is given, and the discrepancy with the recent estimates obtained by Hamer and Peierls, Hwa, and Henyey is explained. Finally, our analysis shows that the random walk picture of the multiperipheral models is not valid for realistic density of particles in rapidity scale. The measurements of the elastic and inelastic scattering cross-sections at NAL and ISR energies have stimulated a new interest in the problem of the Van Hove overlap function [1]. In particular, several authors discussed recently this problem in the framework of the multiperipheral model [ 2 - 4 ] . In the present letter we report briefly the study of the overlap function in two specific versions of the multiperipheral model, namely the Chan-LoskiewiczAllison and Chew-Pignotti models. The more detailed description of the calculations and of the results will be published elsewhere [13]. Our arguments and c o n clusions are based on exact Monte Carlo calculations performed up to energy 1500 GeV, and taking into account energy and momentum conservation. These calculations show that the slope of the overlap function is much smaller than that of elastic amplitude. Thus they agree with earlier calculations at lower energies, but they are in strong disagreement with conclusions of refs. [ 2 - 4 ] . In order to introduce the reader into the problem let us write down the equation for the imaginary part of the two body (elastic) amplitude, which is direct consequence of the S-matrix unitarity.
lm
A2(paPb;pa,Pb,)
(1)
where A 2 is an elastic scattering amplitude a+b ~ a+b,A n is an amplitude for production n particles a+b -* 1+2+ ... +n, and drn is the Lorentz invariant phase space element. An adequate model of particle production inserted into the right-hand side of eq. (1) has to generate correct amplitude of the elastic scattering, which is believed to be dominantly imaginary in the high energy limit. Thus the eq. (1) provides a test of models for particle production. As it has been shown by Michejda and collaborators [5-8] this test is rather severe and many models do not satisfy it. The RHS of eq. (1) is called the overlap function [1 ]. In the following we consider its inelastic part (n > 2) which determines elastic scattering amplitude up to 10%. We analyzed two types of multiperipheral parametrization for the matrix element. One of them was Chan-Loskiewicz-Allison model [9-10] widely studied at energies below 30 GeV. In this model modulus of the matrix element for n particles production is of the form
t +c°
(2)
where
2 = ~ 2 fdrn An(PaPb; Pl 'P2 "'"Pn)A*(Pa'Pb';Pl "'"Pn)
=(Pi +Pi+1 )2 _(mi +mi +1)2,
ti= (Pa-- k=lPk) 369
Volume 50B, number 3
PHYSICS LETTERS
r
5
10 June 1974
7
"-"
4 3
b
,! 2
3
1
lb
2b
3(3 n
Fig. 1. Partial overlap slopes versus total multiplicity In the
CLA modle at Plab: a-50, b-300, c-1500 GeV. Parameters g, c, a, b were taken fromthe original paper [9]. We have performed Monte Carlo calculations of the inelastic overlap function for the whole total multiplicity spectrum at 50, and 300 GeV, and for multiplicities close to the average at 1500 GeV (estimated from logarithmic fit). We have taken only one multiperipheral graph, and only with meson reggeons exchanged (with intercept 0.5). We disregarded all other graphs with permutated positions of the final particles. The phases of the amplitudes were taken constant. In our calculations we used a new method for generating multiperipheral Monte Carlo events, developed for high energy calculations. Detailed description of the method will be published elsewhere. In figs. l and 2 we present the results of our calculations. As we can see the slopes of partial overlap functions are small in comparison with the slope of the elastic amplitude (6 GeV -2) and decrease with increasing multiplicity of intermediate state, contrary to the prediction of the random walk picture (fig. 1). It is seen also that the obtained shrinkage is comparable to the experimental one (fig. 2). Thus our results agree with calculations at lower energies [7]. They disagree entirely, however, with those of refs. [ 2 - 4 ] , in which it was concluded that (i) the slope of the overlap function in the multiperipheral model is bigger than the experimentally determined, and (ii) the shrinkage is much stronger than experimentally observed. As we shall show below, this difference arises from the fact that authors of refs. [2-4] neglected longitudinal part of the four momentum transfer t L. In the 370
loo
2oo
5oo
looo
2ooo . I'c v ]
ooo
Fig. 2. Slope of the overlap function versus In(s) at average total multiplicity, a-experimental, b-from CLA model, c-from CP model.
multiperipheral model the slope of the overlap function is, roughly speaking, proportional to 1/(ti). In the approximation of the vanishing t L
(ti)~.(tT)= -
~pT k k=l
(3)
Taking into account experimental distribution of the transverse momenta of produced particles we get t ~. - 0 . 1 . From our calculations it appears that for the multiplicities close to average, average longitudinal momentum transfer is around - 1 . 6 GeV 2. Thus formula (3) leads to underestimation of the average t by order of magnitude and, in consequence, to overestimation of the overlap function slope and shrinkage. Let us also point out that the average value of the parameter a i = In (1 +si/bi) which governs the momentum transfer distribution in the CLA model has nothin common with the range of nuclear forces if (t i) > (si). In such evidently not multiperipheral situation a i should be regarded as pure phenomenological parameter. Although bounds on four momentum transfers are very weak, transverse momenta are strongly bounded, and we get their average values consistent with the data. To,see how it happens let us write a formula for longitudinal momentum transfer along ith link in multiperipheral chain
Volume 50B, number 3
PHYSICS LETTERS
2
i
2
(4)
10 June 1974
We have performed the same calculations at the same energies and multiplicities for the simple ChewPignotti model n-1
Taking into account energy momentum conservation and assuming that produced particles are equally spaced in rapidity scale with distance d we can derive the following approximate formula t L ~ (mT) 2 e-d/(l_e-d)
2
(5)
where m T = (m 2 +pT 2)1/2. This approximation is valid for links far from the ends of the multiperipheral chain. The approximate transverse momentum dependence of the amplitude through t L is exp (aeffp T) where ~eff <°~)e-a/(1-e-d) 2. Taking d = In s/
IA n I = i~_l f(si)
exp
(ati),
(6)
In order to reproduce the experimental average transverse momenta we had to take for constant a the value about 0.5-0.4 GeV -2 in rough agreement with CLA value (a i) = 0.4, and in strong disagreement with approximation a = 2/<(pT) 2) = 5.5 GeV -2 used in refs. [2-4]. Comparing the results of the Monte Carlo calculations with CLA and CP models at average total multiplicities for different energies (see fig. 2), we can see that there is not substantial difference between them as far as the overlap function and its shrinkage are concerned. We conclude that flatness of the overlap function cannot be attributed to the particular CLA parametrization but (in the framework of the models with single emission from each vertex) is a general feature of the multiperipheral kinematics. To summarize, we have shown by means of the Monte Carlo calculations for pp collisions at Plab 50, 300, and 1500 GeV that in the Chan-LoskiewiczAllison and Chew-Pignotti models without momentum dependent phases slope of the overlap function is four times smaller than it was found experimentally. Shrinkage was found close to experimental value. These results are in disagreement with calculations based on the approximation in which longitudinal momentum transfers are neglected. We proved that this approximation, which is incidentally the basis of the random walk picture, cannot be justified for multiplicities close to and larger than average. Non negligible longitudinal momentum transfers result from the high particle density in the rapidity scale and energy-momentum conservation. The authors would like to thank Professor A. Bialas for his help and encouragement. They thank also Dr. K. Fialkowski and Professor K. Zalewski for readhag of the manuscript and critical remarks.
371
Volume 50B, number 3
PHYSICS LETTERS
References [1] [2] [3] [4] [5] [6] [7]
372
L. Van Hove, Rev. Mod. Phys. 36 (1964). R.C. Hwa, Phys. Rev. D8 (1973) 1331. C.J. Hamer and R.F. Peierls, Phys. Rev. D8 (1973) 1358. F.S. Henyey, Phys. Lett. B45 (1973) 469. L. Michejda, Nucl. Phys. B4 (1968) 113. L. Michejda, Fortschr. Phys. 16 (1968) 707. L. Michejda, J. Turnau and A. Bialas, Nuovo Cimento A56 (1968) 241.
10 June 1974
[8] A. Jurewicz, L. Michejda, J. Namyslowski and J. Turnau, Nucl. Phys. B29 (1971) 269. [9] H.M. Chan, J. Loskiewicz and W.W.M.Allison, Nuovo Cimento A57 (1968) 93. [10] Z. Ajduk, L. Michejda and W. W6jcik, Acta Phys. Pol. A37 (1970) 285. [11] V.N. Gribov, Yad. Fiz. 9 (1969) 640; J.D. Bjorken, preprint SLAC-PUB-905 (1971). [12] G.F. Chew and A. Pignotti, Phys. Rev. 176 (1968) 2112. [13] S. Jadach and J. Turnau, preprint TPJU-3/74, February 1974; Acta Phys. Pol. B, to be published.