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Comparison of production of large mass and of large transverse momentum
Volume 63B, number 3
PHYSICS LETTERS
COMPARISON OF PRODUCTION
2 August 1976
OF LARGE MASS
AND OF LARGE TRANSVERSE
MOMENTUM
C. MICHAEL
Department of Applied Mathematics and Theoretical Physics, University of Liverpool, P.O. Box 147, Liverpool, UK Received 12 July 1976 The production in the central region of particles of different mass Or, K, D, p, ~, ~) and of varying transverse momenta are related using a simple approach to multiparticle production.
Attention is focused on particle production in the central region, particularly at 90 ° (i.e. Yc.m. = 0) where the invariant cross section is largest in general and the lingering effects of beam and target fragmentation are smallest. A comparison is made o f the production of particles of different masses and at different transverse momenta in order to establish a simple relative dependence on these variables for very high energy production. Several classes of models of multiparticle production suggest that the appropriate variable is the particle transverse mass m T = (p2 + m2)1/2 which is equal to its energy E in the frame in which its longitudinal momentum is zero (e.g. the centre of mass frame for production at 90°). A high energy collision can result in the fragmentation of the two incoming particles into fragments with finite x plus a pionization, bremsstrahlung or statistical emission of particles at x = 0. The latter component is damped approximately exponentially in the energy of the particles produced, or more correctly in the transverse mass. The multiperipheral mode is one example that yields such a scheme and the asymptotic behaviour can be analysed using the Mueller-Regge approach. In the central re#on, the double Regge limit is appropriate and it will be dominated by double Pomeron exchange at high energy. This leads naturally to a universal dependence on m T (not necessarily on exponential) with an overall strength proportional to the particle couplings to the Pomeron. Detailed schemes based on f-dominance have been proposed [ 1 ], but the general expectation is that the inclusive spectrum of particle C in protonproton collisions will be given asymptotically by
dy~2~= 0 ~ O20T(Cp)f(mT).
(1)
There may be a very weak residual dependence on the total energy x/~-arising from the deviations from a pure pole of the expressions used for Pomeron exchange, but since the relevant sub-energies are ~ mTX/~-this will be small in practice and anyway it will cancel out in particle production ratios at the same energy. This expression gives an asymptotic estimate of the relative reduction in production cross section of particles of different quark content (~°_ZOT (Cp)) and of different mass (via f ( m T ) = f(x/mZp{)). It relates 7r, 77 and K production at ISR energies successfully [ 1 ]. To use the relation at existing energies for the production of heavier particles or of larger transverse momentum, care must be taken of nonasymptotic corrections. For instance, there is a substantial energy dependence of ~r- production at PT = 3 (GeV units are used throughout) where the yield increases by a factor of 5 from X/s = 23 to x/s-= 53 [2]. A comparison with eq. (1) needs such data to be extrapolated to an energy where the particle yield is approximately energy independent. Consider, for example, proton-proton collisions at PLAB ~ 200 GeV/c (corresponding to x/~-" 20) producing a 7r with m T ~ 3, m "" 0 and PT ~ 3 as against a ff with m T "- 3, m ~ 3 and PT "" 0. In each case m T is the same so that asymptotically one would expect a ratio of O20T(nP)/O2TOT(~p) ~ 100 between the rates. In fact the observed ratio is near one at 200 GeV/c as shown in fig. 1. The energy dependence of PT = 3 production of n - is strong and an estimate of
301
Volume 63B, number 3
PHYSICS LETTERS
PP
P~G'200 G~v/~
nff
io°
".~ E
I0"~
m
"o 10-~
0
i
i
i
i
1
2
3
4
% : (~ +
m')½
GeV
Fig. 1.
the asymptotic production from the fit of the BritishScandinavian group [2] is shown in the figure. On the other hand, do/dy for ~b production is comparable at X/~-~ 17 [2] and X R ~ 53 [5]. This suggests that the asymptotic ratio at m T "- 3 of n to ~ production is more like 50-100 in agreement with the expression ofeq. (1). Sub-asymptotic corrrections to eq. (1) of order s-1/2arising from threshold-like effects have been discussed [6]. In order to understand the more substantial energy dependence observed at larger PT at existing energies, a simple analysis of the constraint of energy-momentum conservation will be presented. This will explain naturally the different energy dependence of n and ff production at m T = 3 from x R -= 20 to higher energies. In either case the same energy (E = roT) has to be produced and this will be balanced (relative to a typical event) by reducing the longitudinal momenta of the other produced particles. The difference lies, however, in the transverse momentum of 3 GeV that must be balanced in the case of 7r production. This requires the other n produced particles to have increased transverse momentum on average and this reduces the cross section. A rough estimate of this effect can be given by considering the large transverse momentum I of the produced 7r to be 302
2 August 1976
balanced by n particles which are uncorrelated in transverse momentum and each have a gaussian de,. pendence on PT with average
Volume 63B, number 3
PHYSICS LETTERS
Thus a realistic account of the constraints of energy momentum conservation explains the suppression at existing energies of large PT production data relative to large mass production. A simple picture with a universal m T dependence at asymptotic energies is thereby supported. It is useful to contrast this with parton model approaches which are valid for finite pT/X/s or m/x/s so that the asymptotic limit is a special case in such models. A specific choice of mechanism yields a PT dependence like p~8 while a Drell-Yan approach yields an m dependence like m - 4 for vector meson production. This gives a more rapid decrease with increasing PT than with increasing mass but with
2 August 1976
no unification. Our approach suggests that asymptotically the PT and m dependences will be the same.
References [11 T. Inami, DESY preprint 75/15 (1975). [2] B. Alper et al., Nucl. Phys. B100 (1975) 237. [3] K.J. Anderson et al., Phys. Rev. Lett. 36 (1976) 237; Proc. Intern. Conf. on Production of particles with new quantum numbers, Madison-Wisconsin, to be published; and private communication. [41 R. Singer et al., Phys. Lett. 60B (1976) 385. [5] F.W. Busser et al., Phys. Lett. 56B (1975) 482. [6] L. Caneschi, Nucl. Phys. B68 (1974) 77. [7] Y.M. Antipov et al., Phys. Lett. 60B (1976) 309.
303
PHYSICS LETTERS
COMPARISON OF PRODUCTION
2 August 1976
OF LARGE MASS
AND OF LARGE TRANSVERSE
MOMENTUM
C. MICHAEL
Department of Applied Mathematics and Theoretical Physics, University of Liverpool, P.O. Box 147, Liverpool, UK Received 12 July 1976 The production in the central region of particles of different mass Or, K, D, p, ~, ~) and of varying transverse momenta are related using a simple approach to multiparticle production.
Attention is focused on particle production in the central region, particularly at 90 ° (i.e. Yc.m. = 0) where the invariant cross section is largest in general and the lingering effects of beam and target fragmentation are smallest. A comparison is made o f the production of particles of different masses and at different transverse momenta in order to establish a simple relative dependence on these variables for very high energy production. Several classes of models of multiparticle production suggest that the appropriate variable is the particle transverse mass m T = (p2 + m2)1/2 which is equal to its energy E in the frame in which its longitudinal momentum is zero (e.g. the centre of mass frame for production at 90°). A high energy collision can result in the fragmentation of the two incoming particles into fragments with finite x plus a pionization, bremsstrahlung or statistical emission of particles at x = 0. The latter component is damped approximately exponentially in the energy of the particles produced, or more correctly in the transverse mass. The multiperipheral mode is one example that yields such a scheme and the asymptotic behaviour can be analysed using the Mueller-Regge approach. In the central re#on, the double Regge limit is appropriate and it will be dominated by double Pomeron exchange at high energy. This leads naturally to a universal dependence on m T (not necessarily on exponential) with an overall strength proportional to the particle couplings to the Pomeron. Detailed schemes based on f-dominance have been proposed [ 1 ], but the general expectation is that the inclusive spectrum of particle C in protonproton collisions will be given asymptotically by
dy~2~= 0 ~ O20T(Cp)f(mT).
(1)
There may be a very weak residual dependence on the total energy x/~-arising from the deviations from a pure pole of the expressions used for Pomeron exchange, but since the relevant sub-energies are ~ mTX/~-this will be small in practice and anyway it will cancel out in particle production ratios at the same energy. This expression gives an asymptotic estimate of the relative reduction in production cross section of particles of different quark content (~°_ZOT (Cp)) and of different mass (via f ( m T ) = f(x/mZp{)). It relates 7r, 77 and K production at ISR energies successfully [ 1 ]. To use the relation at existing energies for the production of heavier particles or of larger transverse momentum, care must be taken of nonasymptotic corrections. For instance, there is a substantial energy dependence of ~r- production at PT = 3 (GeV units are used throughout) where the yield increases by a factor of 5 from X/s = 23 to x/s-= 53 [2]. A comparison with eq. (1) needs such data to be extrapolated to an energy where the particle yield is approximately energy independent. Consider, for example, proton-proton collisions at PLAB ~ 200 GeV/c (corresponding to x/~-" 20) producing a 7r with m T ~ 3, m "" 0 and PT ~ 3 as against a ff with m T "- 3, m ~ 3 and PT "" 0. In each case m T is the same so that asymptotically one would expect a ratio of O20T(nP)/O2TOT(~p) ~ 100 between the rates. In fact the observed ratio is near one at 200 GeV/c as shown in fig. 1. The energy dependence of PT = 3 production of n - is strong and an estimate of
301
Volume 63B, number 3
PHYSICS LETTERS
PP
P~G'200 G~v/~
nff
io°
".~ E
I0"~
m
"o 10-~
0
i
i
i
i
1
2
3
4
% : (~ +
m')½
GeV
Fig. 1.
the asymptotic production from the fit of the BritishScandinavian group [2] is shown in the figure. On the other hand, do/dy for ~b production is comparable at X/~-~ 17 [2] and X R ~ 53 [5]. This suggests that the asymptotic ratio at m T "- 3 of n to ~ production is more like 50-100 in agreement with the expression ofeq. (1). Sub-asymptotic corrrections to eq. (1) of order s-1/2arising from threshold-like effects have been discussed [6]. In order to understand the more substantial energy dependence observed at larger PT at existing energies, a simple analysis of the constraint of energy-momentum conservation will be presented. This will explain naturally the different energy dependence of n and ff production at m T = 3 from x R -= 20 to higher energies. In either case the same energy (E = roT) has to be produced and this will be balanced (relative to a typical event) by reducing the longitudinal momenta of the other produced particles. The difference lies, however, in the transverse momentum of 3 GeV that must be balanced in the case of 7r production. This requires the other n produced particles to have increased transverse momentum on average and this reduces the cross section. A rough estimate of this effect can be given by considering the large transverse momentum I of the produced 7r to be 302
2 August 1976
balanced by n particles which are uncorrelated in transverse momentum and each have a gaussian de,. pendence on PT with average
Volume 63B, number 3
PHYSICS LETTERS
Thus a realistic account of the constraints of energy momentum conservation explains the suppression at existing energies of large PT production data relative to large mass production. A simple picture with a universal m T dependence at asymptotic energies is thereby supported. It is useful to contrast this with parton model approaches which are valid for finite pT/X/s or m/x/s so that the asymptotic limit is a special case in such models. A specific choice of mechanism yields a PT dependence like p~8 while a Drell-Yan approach yields an m dependence like m - 4 for vector meson production. This gives a more rapid decrease with increasing PT than with increasing mass but with
2 August 1976
no unification. Our approach suggests that asymptotically the PT and m dependences will be the same.
References [11 T. Inami, DESY preprint 75/15 (1975). [2] B. Alper et al., Nucl. Phys. B100 (1975) 237. [3] K.J. Anderson et al., Phys. Rev. Lett. 36 (1976) 237; Proc. Intern. Conf. on Production of particles with new quantum numbers, Madison-Wisconsin, to be published; and private communication. [41 R. Singer et al., Phys. Lett. 60B (1976) 385. [5] F.W. Busser et al., Phys. Lett. 56B (1975) 482. [6] L. Caneschi, Nucl. Phys. B68 (1974) 77. [7] Y.M. Antipov et al., Phys. Lett. 60B (1976) 309.
303