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Thermodynamic inclusive single particle spectra and particle production ratios at the ISR
Volume 41B, number 5
PHYSICS LETTERS
30 October 1972
THERMODYNAMIC INCLUSIVE SINGLE PARTICLE SPECTRA AND PARTICLE PRODUCTION RATIOS AT THE ISR J. RANFT* CERNLaboratory H, Geneva,Switzerland Received 21 August 1972 Two essential modifications of the thermodynamic single particle spectra using fireballs according to the strong bootstrap model are described: (1) An energy-dependent velocity weight function is introduced which leads to better agreement with the flat singleparticle spectra in the central region observed at the ISR. (ii) Heavy pair production is modified by producing the two particles in different steps of the decay chain of the fireballs. The fact that heavy fireballs are produced leads in the model to a natural understanding of the deviations from scaling behaviour in the central region of single particle spectra p + p ~ n ±,~ or K- + anything between 20 GeV and ISR energies. The particle producUon ratios ~/~r-, K-Dr-, ~r+/p,K+/p and ff/p are compared to recent ISR data. Inclusive single particle spectra according to the weak thermodynamic bootstrap model proposed by Hagedom in 1965 [1] were first calculated by Hagedorn and Ranft [2] for p - p collisions. More features of these spectra were discussed later [3-6] and the model was also applied to K ° and pion production in Kp collisions [ 7 - 9 ] . The spectra according to the weak thermodynamic bootstrap model were characterized by a limiting fragmentation behaviour [10, 11] and did lead at asymptotic energies s ~ ~ to a vanishing central plateau [5]. In 1971 'the strong thermodynamic bootstrap solution was proposed by Frautschi [ 12] and has since been applied to the calculation of inclusive spectra of n ±, p, ~, K - and 7 produced in p - p collisions [13, 14] and to pion production in ~rp collisions [ 15]. Perhaps the most essential change in the strong bootstrap model which influences inclusive particle spectra is the different decay mechanism of the fireballs. Fireballs of mass MF according to the weak bootstrap solution decay in the first generation to a number n "" l n M F of secondaries, a few of these are expected to decay further in the following generations but no long decay chain is possible. In the strong bootstrap model the fireball decays independently of its mass in one generation into about n ~ 2.4 secondaries, one of these being again a heavy fireball and the other particles are light ones, mainly pions. One estimates easily that the number of steps q(MF) in this * Visitor from Sektion Physik, Karl-Marx-Universit~it.Leipzig, DDR.
linear decay chain is proportional to the original fireball mass MF q(MF) = qoMF.
(1)
This decay chain however does not change significantly the Planck-type momentum spectra fi(E, T) ~ [exp (E'/T) + 1] -1 of the secondaries emitted as has been demonstrated by Monte Carlo calculations [ 16]. In this respect the fireball decay chain differs from the decay chain of novas which is assumed to lead to secondary momentum spectra of Gaussian shape [ 17]. Therefore it is justified to take the fireball decay chain into account by replacing the original Plhnck type spectra by [ 13]
f ' i" ( E ' ' T(MF)' MF) = q(MF)fi(E'"T(MF))
(2)
This leads to significant changes in the behaviour of the spectra in the strong bootstrap model [13] against the model using the weak bootstrap solution. Scaling [ 18] and a non-vanishing flat central plateau with one small central dip (of constant, energy independent width when plotted against the rapidity variable) is found [13]. When the dip was found [13] it was not known whether such a dip would be present at ISR energies but it was discussed that the dip can be removed by introducing a slight energy dependence of the velocity weight function F(X) ~k(')'±--1) + 1 FnewGk, 7±) = Fold(~.) ,~2(7+ _1)2 + 2X(3'± - I ) (3) 613
Volume 41B, number 5
PHYSICS LETTERS
where the velocity parameter of the fireballs is defined as )t = (sign/3)(3,-1)/(%-l)-and where/~ and 7 are the Lorentz parameters of fireballs and 3,_+the Lorentz parameters of the two primary particles m the colhsion. The old velocity weight function was parametrlzed as [131 F(X)
=
N exp (-alX[);
a = 0.76.
(4)
In the meantime more data on particle production at ISR energies have been reported [19] and the impression is that no central dip is present. Therefore it is advisable to compute the thermodynamic spectra w~th the energy-dependent function (3). A more practical nolation is introduced by replacing the integratmn over X in the thermodynamic single particle spectra
fl (P) = E d3N/d3p 1 = fdXF(X,7-+)q(MF(X,',/_+)E'fl(E' , 7)
(5)
J
-1 by the integration over the fireball rapidity r/= arcosh 7. Furthermore one might replace the product of the velocity weight function F(X,7_+) according to (3), the decay chain multiplicity function q(M F (X,?_+)) according to (1) wlthM F (X,V+) as given below in (10) and the Jacobmn dX/dr/by one function of the rapiditles [15] G(r/,r/_+) = F(X,3,+)q(MF Gk,3'±)) dX/dr/.
(6)
Neglecting unessential terms one obtains the form [15]
G(r/,r/._) = N q 0 exp ( - a exp × { 1 - (2m sinh r/)/EcM },
(-Ir/-r/_+l)} (7)
(r/+ and m+ for r / > 0, r/_ and m for 7/< 0) where m_+ are the masses of the initial particles moving with the rapiditles r/+ = arcosh ~/_+.With(6) and (7) the inclusive thermodynamic single particle spectra for particle spectra for particles which can be produced freely without restrictions from conservation laws becomes j~(p)
=
f
--'O
E
d3N/d3p
=
(8)
dr/G(r/,r/~)/a~ cosh (Yi-r/~)Ji ~ T(MF (r/,r/±)) ]
(r/+ for 7/> 0, r/_ for r / < 0), w h e r ~ e rapidity of the produced particle and/a~ = Vp~ + m~ is its longitudinal mass. 614
30 October 1972
In the weak thermodynamic bootstrap heavy pairs (particles 1 and j) like p~ have both to be produced from fireballs in the first generation of decay. This leads to spectra for one of these particles (1) in the form [2] d3Nj
1 - f dX F(X) 2 N I (X) L (X, 7_+) )] (E ', T(X, ~/_+)), (9) d3p -1
where NI(X) is the total multipllmty of particles belonging to family I (with the same quantum numbers as particle i produced together with particle j) resulting from the decay of a fireball moving with X. In the strong bootstrap model the fireballs decay via a long decay chain. It is not very probable and not necessary that particles i and j are produced in one step of the decay chain. The production of one ~ from a fireball with B = 0 leads simultaneously to a remaining fireball ofB = 1 which will emit one baryon in a later step of the decay chain. If fireballs with exotic quantum numbers do not exist it is however not possible to emit say two ~ in succession without producing one baryon in between. If exotic fireballs are excluded, the estimations of the total cross sections for heavy pair production by Hagedorn [201 will therefore not be changed essentially by the strong bootstrap model. In ref. [13] the production of heavy particles like ~"and K - was treated as in the weak bootstrap model, adding only the factor q2(X,3,_+) to (9). This did lead to considerable difficulties in explaimng the ~ and K spectra. It was necessary to consider at 20 GeV/c and even 70 GeV/c threshold effects in the ~ and K - spectra which were expected to die out at ISR energies. This made the prediction of these spectra at intermediate energies difficult and unreliable. All these difficulties are removed if we take the strong bootstrap decay chain seriously also for heavy parr production and calculate as outlined above the K - and ~ production without considering the production of an antipamcle from the same generation. We calculate therefore these spectra essentially according to expression (8). For moderate fireball masses we take additionally into account the effective shortening of the decay chain which can lead to the particle of interest. Thi~ is done approximately by subtracting the mass of the second particle i from the original fireball mas M~ = M F - m 1 which is used to calculate q(MF) and T(MF) in (8). To calculate the temperature function T(M F (r/,rl+_))
Volume 41B, number 5
PHYSICS LETTERS
in (8) we have to specify the dependence of the average fireball massMF on 77 or X and the primary energy or 3'±- This dependence influences also the decay chain multiplicity function q(MF) but a change of q(MF) is compensated by a corresponding change of F(~,7±). If the fireballs moving with ;~ become less massive energy conservation leads to an increase of the number of fireballs which compensates the change of q(MF). The temperature T(MF) leads to different particle spectra for different choices OfMF(~,7± ) but this vanation is rather weak. The change in the transverse momentum dependence of the spectra caused by a change OfMF (X,7±) via the temperature function T(MF) is small and might be difficult to detect from data with present experimental errors. Furthermore also presently unknown changes in the shape of the momentum spectra due to the decay chain when treated in the phase space formulation of Frautschi [12] might be of similar order of magnitude. Another way to test the dependence o f MF (;~ = 00'±) from the primary energy using single particle spectra is to study the deviations from scaling behaviour of the spectra in the central region. This will be discussed below, but again this test is rather insensitive with the presently available data. The most promising way to determine MF CA/r±)is presently to study correlations in inclusive two and many particle production [21 ]. All calculations of single particle spectra reported here are done with the function MF (X,7±) = TEeM -- 4E2M(72 --l) + m±2
(10)
which gives the maximum kinematically possible mass MF [22], but we stress again that changes by a factor of about 2 will not lead to significantly different predictions. The spectra of ~ and K - and the dominant contribution to the 7r- spectra in the central region are calculated according to formula (8). There is one free parameter a which appears in the velocity weight function and in the rapidity function G(r/,rh). This parameter characterizes the velocity distribution of produced fireballs and is therefore the same for all kinds of produced particles. The normalization factor q0 of the decay chain multiplicity function ( 1) which appears also in G (r/, r/±) has to be determined from experiment. This parameter is expected to differ slightly for secondary particles of different mass because of the different length of the ef-
30 October 1972
fective decay chains. These four parameters have been determined from a fit to data keeping the parameters related to the effective interaction volume V and the highest temperature TO as in ref. [13]. The data used in the fit are inclusive n-, K - a n d ~ spectra in pp collisions at 19.2, 24 and 30 GeV/c [23-25] and particle production ratios ~-[rr- and K - / l r - from Serpukhov [26] but no ISR data. The following parameters were obtained a = - 2 . 7 7 , q0(rr-)=3.30, q 0 ( K - ) = 1.40 q0(P) = 1.36
(11)
A smaller effective volume for K production [2] VK = 0.6 F~ has been considered when calculating K spectra and a statistical weight factor 2,/+ 1 = 2 when calculating g spectra. The parameters q0 for K - a n d are smaller than for lr- as expected. The invariant single particle distributions of lr-, K and ~ at five primary momenta between 20 and 1500 GeV/c are compared in fig. 1. In the central region deviations from scaling are found in all cases. These deviations increase with decreasing x =/~/Pllmax and increasing p2. The 7r- spectra at x = 0 increase from 20 to 1500 GeV/c by factors of 1.6 and 2.5 forp 2t = 0 and p~ = 0.4 (GeV/c) 2, respectwely. This agrees closely to the corresponding increase found experimentally [19, 28, 30]. These factors of increase are 3.2 and 5.0 for K-and 14 and 20 for~. The experimental information on K - and ~ production at the ISR results mainly from ratios ~[Tr- and K-/Tr- which were defined [27] using the noninvariant distributions d3Np/d3p
(12)
R(~-_) - d3NJd3p These ratios are plotted in fig. 2 as function ofs and as function o f x for different primary momenta. The data at low energy were used in the fit, we compare the model in fig. 2 with ISR measurements [27, 28] and find good agreement. The ratio ~/Tr- according to the model seems to rise slightly faster with s than the data. We have to consider that the data are averages over transverse momenta whereas the model predicts a change of the ratios with p2. Therefore no quantitative agreement can be expected presently. However if the slower increase of the experimental data should be confirmed this could be used as an indication that the average fireball mass 615
Volume 41B, n u m b e r 5
PHYSICS LETTERS
J---'lSO0 C-eV/c 10 ~_~------250 ,, ~ ,~, ~ 20 ,, ~-'~
.Isoo oev~c
rc-
30 October 1972
. . . . . r,
1 ~1500
\
)v
1
GeV/c
so c,ev~:~-,,.\'%
/-1500 ~ _/-/.500 ~ j l 0 0 ~--%,~f/50
C-eV/c ,, ,, "
~
\\\
-%,~
o,,=o 10"
~o ,, /
_..~.'~
I
10"
\
~o c . , . v , : ~ \
~ r~ 02
"
\
~o ,, _._f___1_._.xx-~. \ ,,
\\
I 2[ 10"0
~ 0t,
~
o,,_-0
,,
uJ 10-1
I
i~
06
10-:
= '~ 02
0
20 C-eVt~-1""~
N
~,
. . . . . . . -,\
\.
".
, ~ 04
\~
06
10 "i
~ 02
0
[ 04
06
X= PpI P~I,rnox
Fig. 1. I n v a n a n t single particle distribution (a) p+p--+Tr-+ a n y t h i n g , (b) p+p--* K - + anything and (c) p + p o ~ + anything for 2 7 primary p r o t o n m o m e n t a Po between 20 and 1500 GeV/c and for transverse m o m e n t a Pt = 0 and 0 4 (GeV/c) plotted as function o f x = max" Deviations from scahng behavlour are f o u n d In the central region x ~- 0 In the model the asymptotic curves are approached from below in the same sense as f o u n d experimentally [19]
pu/pll
o Bert=n et am
,+
oil- T . _ A ~ ,
, 5 0 0 GeVtc
01
Q
T= 0 01
p~ --U ~ tuev~:}
~
l
t~n el al
'01
•
'°'l~u 0 01
1500 GeV/c
I(3(}~
"6 ty"
o CERN- Allaby et al O Serpuchow-B=non et al • [SR -Albrow et al
~
50 GeV/c
x=035 0 001
0
I
I 04
02 X- Pll Plt,rn~
I 06
~
)01 10
l
I
100
1000
10000
s (GeV 2)
Fig. 2 Particle production ratios K - / r r - and ~[rr- for primary p r o t o n m o m e n t a Po between 20 and 1500 GeV/c; (a) Plotted as funct]on o f x = m a x for p~ = 0.4 (GeV/c) 2. The curves according to t h e model are compared to ISR m e a s u r e m e n t s by Albrow et al. [27] and Bertin et al. [ 2 8 ] . The experimental points are averaged over 0.2 < Pt < 0.9 GeV/c. Particle production data at 20 to 70 GeV have been used m the fit and agree with the curves; (b) Plotted as function o f s (GeV) 2 for x = 0.35 and compared to data from CERN [23,241, Serpukhov [261 and the ISR [27]
Pll/Pll
616
Volume 41B, number 5
PHYSICS LETTERS
~ P
10
The model
( p~;o),P~=~
X
•
ISR-AI~ow et
c~ al.
30 October 1972
pected but mainly by a further decrease of_the leading proton spectra in the central region. In fig. 3 we present a comparison of measured ratios R(lr+/p) and R(K+/p) [27] with the predictions o f the model and find also good agreement. Useful discussions and suggestions by R. Hagedorn and J.C. Sens are acknowledged.
p
References Q
10-
i0 -2
0
I 02
06
06
x_-91 ~11)max
Fig. 3. Particle ~roduction ratios ~+/p and K+/p according to the model for Pt -- 0 compared to ISR data [27] measured at 0.2 < Pt < 1.1 GeV/c. The data at 20 to 30 GeV/c were used in the fit reported m ref. [13]. M F (~,r/+) given in (10) rises too fast with s. The same information can however be extracted more directly from two particle correlations [21]. It should be stress-
ed that we understand the deviations from scaling in the central region easily from our model where clusters or fireballs are formed whose masses increase with primary energy. These deviations could not be understood from a model with constant fireball mass. It might therefore be concluded that the number o f fireballs does not grow strongly with energy. The rr-, K - and p spectra in fig. 1 have nearly reached their asymptotic level also in the central region and no significant further increase is expected with energy. The ratio o f ~ / p production at x = 0 is approximately R ( ~ / p ) = 0.5 [19]. The same ratio is found in the model. A further approach of this ratio to R (~/p) ~ 1 is ex-
[1] R. Hagedorn. Suppl. Nuovo Cimento 3 (1965) 147. [2] R. Hagedorn and J. Ranft, Suppl. Nuovo Cimento 6 (1968) 169. [3] J. Ranft, Phys. Lett. 33B (1970) 481. [4] J Ranft, Phys. Lett. 31B (1970) 529; [5] J. Ranft, Phys. Lett. 36B (1971) 225. [6] R Blutner et al., Nucl. Phys. B35 (1971) 503 [7] E. Matthaus and J. Ranft, Phys. Lett. 40B (1972) 230. [8] E. Matthaus, Dlplomarbelt Leipzig, unpublished (1972). [9] B. Buschbeck and J. Hodi, Vienna preprmt HEP-II (1972); Nucl. Phys. B, to be published. [10] J. Benecke et al., Phys. Rev. 188 (1969) 2159. [11] R Hagedorn, Nuci. Phys. B24 (1970) 93. [121 S.C. Frautschi, Phys. Rev. D3 (1971) 2821 [13] R. Hagedorn and J. Ranft, CERN preprint TH-1440 (1972); Nucl. Phys. B, to be published. [14] Htun Than et al., Karl-Marx-Universltat Leipzig prepnnt KMU-HEP-7206 (1972). Nuovo CLmento Lett., to be pubhshed. [15] Htun Than and J. Ranft, Kad-Marx-Universitat Leipzig preprint KMU-HEP-7207 (1972), Nuovo Clmento Lett , to be pubhshed. [ 16] J. Kripfganz and H.J. Mohnng, unpublished calculations, Leipztg (1972). [17] M. Jacob and R. Slansky, Phys. Rev. D5 (1972) 1847. [18] R.P. Feynman, Phys. Rev. Lett. 23 (1969) 1415. [ 19 ] J.C. Sens, Invited paper presented at the Oxford Conf, April 1972. [20] R. Hagedorn, Suppl. Nuovo Ctmento 6 (1968) 311. [21] G. Ranft and J. Ranft, CERN preprmt TH 1532 (1972). [22] R. Hagedorn and J. Ranft, CERN preprmt TH 1439 (1972) [23] J.V Allaby et al., CERN report CERN 70-12 (1970) [24] J.V. Allaby et al., Contribution Oxford Conf., April 1972. [25] E.W. Anderson et al., Phys. Rev. Lett. 19 (1967) 198 [26] Yu B. Bushnin et al., Phys. Lett. 29B (1969) 48. [27] M.G. Albrow et al., Phys. Lett. 40B (1972) 136 [28] A. Bertin et al., Phys. Lett. 38B (1972) 260, and paper presented at Oxford Conf., April 1972. [291 L.G. Ratner et al., Phys. Rev. Lett. 27 (1972) 68, and Proc. APD/DPF 1971 meeting at Rochester. [30] British-Scandinavian ISR Collaboration, paper presented by E. Lillethun at Oxford Conf., April 1972.
617
PHYSICS LETTERS
30 October 1972
THERMODYNAMIC INCLUSIVE SINGLE PARTICLE SPECTRA AND PARTICLE PRODUCTION RATIOS AT THE ISR J. RANFT* CERNLaboratory H, Geneva,Switzerland Received 21 August 1972 Two essential modifications of the thermodynamic single particle spectra using fireballs according to the strong bootstrap model are described: (1) An energy-dependent velocity weight function is introduced which leads to better agreement with the flat singleparticle spectra in the central region observed at the ISR. (ii) Heavy pair production is modified by producing the two particles in different steps of the decay chain of the fireballs. The fact that heavy fireballs are produced leads in the model to a natural understanding of the deviations from scaling behaviour in the central region of single particle spectra p + p ~ n ±,~ or K- + anything between 20 GeV and ISR energies. The particle producUon ratios ~/~r-, K-Dr-, ~r+/p,K+/p and ff/p are compared to recent ISR data. Inclusive single particle spectra according to the weak thermodynamic bootstrap model proposed by Hagedom in 1965 [1] were first calculated by Hagedorn and Ranft [2] for p - p collisions. More features of these spectra were discussed later [3-6] and the model was also applied to K ° and pion production in Kp collisions [ 7 - 9 ] . The spectra according to the weak thermodynamic bootstrap model were characterized by a limiting fragmentation behaviour [10, 11] and did lead at asymptotic energies s ~ ~ to a vanishing central plateau [5]. In 1971 'the strong thermodynamic bootstrap solution was proposed by Frautschi [ 12] and has since been applied to the calculation of inclusive spectra of n ±, p, ~, K - and 7 produced in p - p collisions [13, 14] and to pion production in ~rp collisions [ 15]. Perhaps the most essential change in the strong bootstrap model which influences inclusive particle spectra is the different decay mechanism of the fireballs. Fireballs of mass MF according to the weak bootstrap solution decay in the first generation to a number n "" l n M F of secondaries, a few of these are expected to decay further in the following generations but no long decay chain is possible. In the strong bootstrap model the fireball decays independently of its mass in one generation into about n ~ 2.4 secondaries, one of these being again a heavy fireball and the other particles are light ones, mainly pions. One estimates easily that the number of steps q(MF) in this * Visitor from Sektion Physik, Karl-Marx-Universit~it.Leipzig, DDR.
linear decay chain is proportional to the original fireball mass MF q(MF) = qoMF.
(1)
This decay chain however does not change significantly the Planck-type momentum spectra fi(E, T) ~ [exp (E'/T) + 1] -1 of the secondaries emitted as has been demonstrated by Monte Carlo calculations [ 16]. In this respect the fireball decay chain differs from the decay chain of novas which is assumed to lead to secondary momentum spectra of Gaussian shape [ 17]. Therefore it is justified to take the fireball decay chain into account by replacing the original Plhnck type spectra by [ 13]
f ' i" ( E ' ' T(MF)' MF) = q(MF)fi(E'"T(MF))
(2)
This leads to significant changes in the behaviour of the spectra in the strong bootstrap model [13] against the model using the weak bootstrap solution. Scaling [ 18] and a non-vanishing flat central plateau with one small central dip (of constant, energy independent width when plotted against the rapidity variable) is found [13]. When the dip was found [13] it was not known whether such a dip would be present at ISR energies but it was discussed that the dip can be removed by introducing a slight energy dependence of the velocity weight function F(X) ~k(')'±--1) + 1 FnewGk, 7±) = Fold(~.) ,~2(7+ _1)2 + 2X(3'± - I ) (3) 613
Volume 41B, number 5
PHYSICS LETTERS
where the velocity parameter of the fireballs is defined as )t = (sign/3)(3,-1)/(%-l)-and where/~ and 7 are the Lorentz parameters of fireballs and 3,_+the Lorentz parameters of the two primary particles m the colhsion. The old velocity weight function was parametrlzed as [131 F(X)
=
N exp (-alX[);
a = 0.76.
(4)
In the meantime more data on particle production at ISR energies have been reported [19] and the impression is that no central dip is present. Therefore it is advisable to compute the thermodynamic spectra w~th the energy-dependent function (3). A more practical nolation is introduced by replacing the integratmn over X in the thermodynamic single particle spectra
fl (P) = E d3N/d3p 1 = fdXF(X,7-+)q(MF(X,',/_+)E'fl(E' , 7)
(5)
J
-1 by the integration over the fireball rapidity r/= arcosh 7. Furthermore one might replace the product of the velocity weight function F(X,7_+) according to (3), the decay chain multiplicity function q(M F (X,?_+)) according to (1) wlthM F (X,V+) as given below in (10) and the Jacobmn dX/dr/by one function of the rapiditles [15] G(r/,r/_+) = F(X,3,+)q(MF Gk,3'±)) dX/dr/.
(6)
Neglecting unessential terms one obtains the form [15]
G(r/,r/._) = N q 0 exp ( - a exp × { 1 - (2m sinh r/)/EcM },
(-Ir/-r/_+l)} (7)
(r/+ and m+ for r / > 0, r/_ and m for 7/< 0) where m_+ are the masses of the initial particles moving with the rapiditles r/+ = arcosh ~/_+.With(6) and (7) the inclusive thermodynamic single particle spectra for particle spectra for particles which can be produced freely without restrictions from conservation laws becomes j~(p)
=
f
--'O
E
d3N/d3p
=
(8)
dr/G(r/,r/~)/a~ cosh (Yi-r/~)Ji ~ T(MF (r/,r/±)) ]
(r/+ for 7/> 0, r/_ for r / < 0), w h e r ~ e rapidity of the produced particle and/a~ = Vp~ + m~ is its longitudinal mass. 614
30 October 1972
In the weak thermodynamic bootstrap heavy pairs (particles 1 and j) like p~ have both to be produced from fireballs in the first generation of decay. This leads to spectra for one of these particles (1) in the form [2] d3Nj
1 - f dX F(X) 2 N I (X) L (X, 7_+) )] (E ', T(X, ~/_+)), (9) d3p -1
where NI(X) is the total multipllmty of particles belonging to family I (with the same quantum numbers as particle i produced together with particle j) resulting from the decay of a fireball moving with X. In the strong bootstrap model the fireballs decay via a long decay chain. It is not very probable and not necessary that particles i and j are produced in one step of the decay chain. The production of one ~ from a fireball with B = 0 leads simultaneously to a remaining fireball ofB = 1 which will emit one baryon in a later step of the decay chain. If fireballs with exotic quantum numbers do not exist it is however not possible to emit say two ~ in succession without producing one baryon in between. If exotic fireballs are excluded, the estimations of the total cross sections for heavy pair production by Hagedorn [201 will therefore not be changed essentially by the strong bootstrap model. In ref. [13] the production of heavy particles like ~"and K - was treated as in the weak bootstrap model, adding only the factor q2(X,3,_+) to (9). This did lead to considerable difficulties in explaimng the ~ and K spectra. It was necessary to consider at 20 GeV/c and even 70 GeV/c threshold effects in the ~ and K - spectra which were expected to die out at ISR energies. This made the prediction of these spectra at intermediate energies difficult and unreliable. All these difficulties are removed if we take the strong bootstrap decay chain seriously also for heavy parr production and calculate as outlined above the K - and ~ production without considering the production of an antipamcle from the same generation. We calculate therefore these spectra essentially according to expression (8). For moderate fireball masses we take additionally into account the effective shortening of the decay chain which can lead to the particle of interest. Thi~ is done approximately by subtracting the mass of the second particle i from the original fireball mas M~ = M F - m 1 which is used to calculate q(MF) and T(MF) in (8). To calculate the temperature function T(M F (r/,rl+_))
Volume 41B, number 5
PHYSICS LETTERS
in (8) we have to specify the dependence of the average fireball massMF on 77 or X and the primary energy or 3'±- This dependence influences also the decay chain multiplicity function q(MF) but a change of q(MF) is compensated by a corresponding change of F(~,7±). If the fireballs moving with ;~ become less massive energy conservation leads to an increase of the number of fireballs which compensates the change of q(MF). The temperature T(MF) leads to different particle spectra for different choices OfMF(~,7± ) but this vanation is rather weak. The change in the transverse momentum dependence of the spectra caused by a change OfMF (X,7±) via the temperature function T(MF) is small and might be difficult to detect from data with present experimental errors. Furthermore also presently unknown changes in the shape of the momentum spectra due to the decay chain when treated in the phase space formulation of Frautschi [12] might be of similar order of magnitude. Another way to test the dependence o f MF (;~ = 00'±) from the primary energy using single particle spectra is to study the deviations from scaling behaviour of the spectra in the central region. This will be discussed below, but again this test is rather insensitive with the presently available data. The most promising way to determine MF CA/r±)is presently to study correlations in inclusive two and many particle production [21 ]. All calculations of single particle spectra reported here are done with the function MF (X,7±) = TEeM -- 4E2M(72 --l) + m±2
(10)
which gives the maximum kinematically possible mass MF [22], but we stress again that changes by a factor of about 2 will not lead to significantly different predictions. The spectra of ~ and K - and the dominant contribution to the 7r- spectra in the central region are calculated according to formula (8). There is one free parameter a which appears in the velocity weight function and in the rapidity function G(r/,rh). This parameter characterizes the velocity distribution of produced fireballs and is therefore the same for all kinds of produced particles. The normalization factor q0 of the decay chain multiplicity function ( 1) which appears also in G (r/, r/±) has to be determined from experiment. This parameter is expected to differ slightly for secondary particles of different mass because of the different length of the ef-
30 October 1972
fective decay chains. These four parameters have been determined from a fit to data keeping the parameters related to the effective interaction volume V and the highest temperature TO as in ref. [13]. The data used in the fit are inclusive n-, K - a n d ~ spectra in pp collisions at 19.2, 24 and 30 GeV/c [23-25] and particle production ratios ~-[rr- and K - / l r - from Serpukhov [26] but no ISR data. The following parameters were obtained a = - 2 . 7 7 , q0(rr-)=3.30, q 0 ( K - ) = 1.40 q0(P) = 1.36
(11)
A smaller effective volume for K production [2] VK = 0.6 F~ has been considered when calculating K spectra and a statistical weight factor 2,/+ 1 = 2 when calculating g spectra. The parameters q0 for K - a n d are smaller than for lr- as expected. The invariant single particle distributions of lr-, K and ~ at five primary momenta between 20 and 1500 GeV/c are compared in fig. 1. In the central region deviations from scaling are found in all cases. These deviations increase with decreasing x =/~/Pllmax and increasing p2. The 7r- spectra at x = 0 increase from 20 to 1500 GeV/c by factors of 1.6 and 2.5 forp 2t = 0 and p~ = 0.4 (GeV/c) 2, respectwely. This agrees closely to the corresponding increase found experimentally [19, 28, 30]. These factors of increase are 3.2 and 5.0 for K-and 14 and 20 for~. The experimental information on K - and ~ production at the ISR results mainly from ratios ~[Tr- and K-/Tr- which were defined [27] using the noninvariant distributions d3Np/d3p
(12)
R(~-_) - d3NJd3p These ratios are plotted in fig. 2 as function ofs and as function o f x for different primary momenta. The data at low energy were used in the fit, we compare the model in fig. 2 with ISR measurements [27, 28] and find good agreement. The ratio ~/Tr- according to the model seems to rise slightly faster with s than the data. We have to consider that the data are averages over transverse momenta whereas the model predicts a change of the ratios with p2. Therefore no quantitative agreement can be expected presently. However if the slower increase of the experimental data should be confirmed this could be used as an indication that the average fireball mass 615
Volume 41B, n u m b e r 5
PHYSICS LETTERS
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30 October 1972
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Fig. 1. I n v a n a n t single particle distribution (a) p+p--+Tr-+ a n y t h i n g , (b) p+p--* K - + anything and (c) p + p o ~ + anything for 2 7 primary p r o t o n m o m e n t a Po between 20 and 1500 GeV/c and for transverse m o m e n t a Pt = 0 and 0 4 (GeV/c) plotted as function o f x = max" Deviations from scahng behavlour are f o u n d In the central region x ~- 0 In the model the asymptotic curves are approached from below in the same sense as f o u n d experimentally [19]
pu/pll
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0
I
I 04
02 X- Pll Plt,rn~
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100
1000
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Fig. 2 Particle production ratios K - / r r - and ~[rr- for primary p r o t o n m o m e n t a Po between 20 and 1500 GeV/c; (a) Plotted as funct]on o f x = m a x for p~ = 0.4 (GeV/c) 2. The curves according to t h e model are compared to ISR m e a s u r e m e n t s by Albrow et al. [27] and Bertin et al. [ 2 8 ] . The experimental points are averaged over 0.2 < Pt < 0.9 GeV/c. Particle production data at 20 to 70 GeV have been used m the fit and agree with the curves; (b) Plotted as function o f s (GeV) 2 for x = 0.35 and compared to data from CERN [23,241, Serpukhov [261 and the ISR [27]
Pll/Pll
616
Volume 41B, number 5
PHYSICS LETTERS
~ P
10
The model
( p~;o),P~=~
X
•
ISR-AI~ow et
c~ al.
30 October 1972
pected but mainly by a further decrease of_the leading proton spectra in the central region. In fig. 3 we present a comparison of measured ratios R(lr+/p) and R(K+/p) [27] with the predictions o f the model and find also good agreement. Useful discussions and suggestions by R. Hagedorn and J.C. Sens are acknowledged.
p
References Q
10-
i0 -2
0
I 02
06
06
x_-91 ~11)max
Fig. 3. Particle ~roduction ratios ~+/p and K+/p according to the model for Pt -- 0 compared to ISR data [27] measured at 0.2 < Pt < 1.1 GeV/c. The data at 20 to 30 GeV/c were used in the fit reported m ref. [13]. M F (~,r/+) given in (10) rises too fast with s. The same information can however be extracted more directly from two particle correlations [21]. It should be stress-
ed that we understand the deviations from scaling in the central region easily from our model where clusters or fireballs are formed whose masses increase with primary energy. These deviations could not be understood from a model with constant fireball mass. It might therefore be concluded that the number o f fireballs does not grow strongly with energy. The rr-, K - and p spectra in fig. 1 have nearly reached their asymptotic level also in the central region and no significant further increase is expected with energy. The ratio o f ~ / p production at x = 0 is approximately R ( ~ / p ) = 0.5 [19]. The same ratio is found in the model. A further approach of this ratio to R (~/p) ~ 1 is ex-
[1] R. Hagedorn. Suppl. Nuovo Cimento 3 (1965) 147. [2] R. Hagedorn and J. Ranft, Suppl. Nuovo Cimento 6 (1968) 169. [3] J. Ranft, Phys. Lett. 33B (1970) 481. [4] J Ranft, Phys. Lett. 31B (1970) 529; [5] J. Ranft, Phys. Lett. 36B (1971) 225. [6] R Blutner et al., Nucl. Phys. B35 (1971) 503 [7] E. Matthaus and J. Ranft, Phys. Lett. 40B (1972) 230. [8] E. Matthaus, Dlplomarbelt Leipzig, unpublished (1972). [9] B. Buschbeck and J. Hodi, Vienna preprmt HEP-II (1972); Nucl. Phys. B, to be published. [10] J. Benecke et al., Phys. Rev. 188 (1969) 2159. [11] R Hagedorn, Nuci. Phys. B24 (1970) 93. [121 S.C. Frautschi, Phys. Rev. D3 (1971) 2821 [13] R. Hagedorn and J. Ranft, CERN preprint TH-1440 (1972); Nucl. Phys. B, to be published. [14] Htun Than et al., Karl-Marx-Universltat Leipzig prepnnt KMU-HEP-7206 (1972). Nuovo CLmento Lett., to be pubhshed. [15] Htun Than and J. Ranft, Kad-Marx-Universitat Leipzig preprint KMU-HEP-7207 (1972), Nuovo Clmento Lett , to be pubhshed. [ 16] J. Kripfganz and H.J. Mohnng, unpublished calculations, Leipztg (1972). [17] M. Jacob and R. Slansky, Phys. Rev. D5 (1972) 1847. [18] R.P. Feynman, Phys. Rev. Lett. 23 (1969) 1415. [ 19 ] J.C. Sens, Invited paper presented at the Oxford Conf, April 1972. [20] R. Hagedorn, Suppl. Nuovo Ctmento 6 (1968) 311. [21] G. Ranft and J. Ranft, CERN preprmt TH 1532 (1972). [22] R. Hagedorn and J. Ranft, CERN preprmt TH 1439 (1972) [23] J.V Allaby et al., CERN report CERN 70-12 (1970) [24] J.V. Allaby et al., Contribution Oxford Conf., April 1972. [25] E.W. Anderson et al., Phys. Rev. Lett. 19 (1967) 198 [26] Yu B. Bushnin et al., Phys. Lett. 29B (1969) 48. [27] M.G. Albrow et al., Phys. Lett. 40B (1972) 136 [28] A. Bertin et al., Phys. Lett. 38B (1972) 260, and paper presented at Oxford Conf., April 1972. [291 L.G. Ratner et al., Phys. Rev. Lett. 27 (1972) 68, and Proc. APD/DPF 1971 meeting at Rochester. [30] British-Scandinavian ISR Collaboration, paper presented by E. Lillethun at Oxford Conf., April 1972.
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