Scaling method for super-high energies from experimental data below 30 GeV

NUCLEAR INSTRUMENTS

A N D M E T H O D S IO3 (1972) 555-564; ',~b N O R 1 H - H O L L A N D

PUBLISHING

CO.

SCALING M E T H O D FOR S U P E R - H I G H E N E R G I E S F R O M E X P E R I M E N T A L DATA B E L O W 30 GeV G. CECCHET, R. DOLFINI, F. IMPELLIZZERI and S RATTI lstttutto di Fistca and Sezione INFN, Mdan, Italy Received 17 Aprd 1972 A general method, not of the Monte Carlo type, ~s proposed to s,mulate elementary events at very high energ,es Actual experimental data obtained at energies below 30 GeV are scaled up under different scahng hypotheses. The bastc prmoples are d~scussed and several distributions in the laboratory system are gwen for the reaction at 65, 100, 150, 200 GeV.

c. to provide a realistic sample of events of a given channel, suitable for estimating detection efficiencies, experimental resolutions, etc., for a particular detector (BEBC, Omega magnet, particular experimental set-ups, etc.). In section 2 we outline the general principles, and in sections 3 and 4 we sketch two alternative versions of the scaling method. In section 5 we discuss and compare the resulting high energy distributions.

1. Introduction In high energy strong interactions a considerable amount of data has been collected by different laboratories at energies below 30 GeVI). Nonetheless the information on multiparticle reactions has a nature which practically forbids a detailed compilation of physical quantities, the whole physical content being included only in the knowledge of the four-momenta of all final particles2). In this respect only a complete Data Summary Tape (hereafter DST) summarizes all the physical informations. On the other hand DST on particular exclusive channels at a given energy (or in some cases at several energies) are already available. Typical examples are the World Collections of Data on the reactions p p ~ p p n + 7z- ,

(1)

n-p--, ~r- p~+ 7r- ,

(2)

K+ p--,K+ pTt + rc - .

(3)

2. Basic principles Recently, two characteristics of high energy hadron processes have become evident 3): a. Scaling behaviour holds approximately for the limiting one-particle distribution in the asymptotic energy region ; b. The accelerator energies actually available (say s > 20 GeV 2) are not far from the asymptotic region4). Under these two assumptions let us consider a general scattering process of the type

The analysis of the trend shown by specific properties of these channels, as a function of the incoming momentum, suggests a method suitable to provide a realistic prediction of the kinematical configurations which the same reactions might show at energies of the order of 100 GeV, which will be available at the accelerators of the new generation. The purpose of this paper is threefold: a. to propose a general method, which is not of the Monte Carlo type, useful to estimate, in a way as realistic as possible, a set of four-momenta for the given final particles at an energy ~/s2, given the knowledge of their four-momenta at an energy X/sl, already experimentally studied; b. to provide estimated distributions (angular distributions, momentum distributions, etc.) at energies presumably available at N A L and C E R N 2;

a + b - - * y. c,,

(4)

l=l

sketched in fig. i. The notations (in the cms) are:

f(s, q,, r,, m,) scattering amplitude,

555

s square of the total energy, q~ longttudinal momentum of the ith final particle,

L $ P

~

¢

Fig. 1.

2

556

G. CECCHET

r, transverse m o m e n t u m of the ith final pamcle, m, mass of the ith final particle. It is shown that at least approximately 5) one can write:

f(s, q,, r,, nh) ~. p(s, q,) g(r,, m~).

et

al.

where 2q, X~

N/SI

2Q, (5)

X~

$~oc

Alternatwely one can use instead of q, the "reduced variables ''6)

x/s2

The simplest way to verify identity (12) is: X~ = x,.

2q, ~,[q,]

x,

x,

or

v/s

(6)

and use

f'(s, x,, r,, m,) ~ p'(s, x,) g(r,, mi).

(5')

F r o m eq. (5) or (5') it is possible to make a "minim u m " set of hypotheses in order to generate one event at an energy ~,/se for each real event measured at a given energy V/s,. T o be more specific let us consider an exclusive reaction. F o u r - m o m e n t u m conservation gives y. q, = 0,

or

y. x, = 0,

t

(7)

t

y , , = 0,

(8)

~t

E, = ,,/sl.

F r o m eqs. (l 1) and (12) energy conservation eq. (9) is not satisfied. The consequence of this is that, in practice, the more peripheral is the configuration of the event, the better is the approximation in the energy conservation. By reversing the argument, by imposing energy conservation, from eqs. (11) and (l 3) generates events at slightly different ~,/s2*. 2. q, is "scaled u p " in an hypothesis very near to a " p u r e fragmentation hypothesis" (two fireballs). In this case one can assume"

p(s2, Q,) : p(sl + e 2, q , + G),

Now: a. Scaling of g(r,, m,) may suggest that a "generated event" maintains for each final particle at X/s2 the same transverse m o m e n t u m r, measured at the energy X,'sl i.e.

R, =r,,

(10)

Q, = q , + G .

x/s2.

Energy and m o m e n t u m conservation puts limitations on the G's. F r o m eqs. (15) and (7): y. •, = 0.

(16)

F r o m eq. (9)

•[(ql+K,)2+r2+m2]

½ ----x/s2,

(17)

or in the asymptotic limit (11)

l

[q, +Kil ~ x//s2.

(17')

t

b. A proper scaling ofp(s, q~) has to be found in order to "scale u p " Q, at ~,/s2 starting from each q, at X 'sx and simultaneously to satisfy the conservation laws at X/s2. T w o simple hypotheses can be made: I. ~, distributions are " f r o z e n " and energy independent as a " p u r e pionization hypothesis" would predict. This means: p(s2, X,) = p(sl, x,),

(15)

i

where capital R, refers to X/s: energy. This automatically verifies at

(14)

where G are parameters to be fixed in order to satisfy eqs. (7) and (9), while e.2 is the difference in s: i.e. s2 = s l + e 2. This can be done by imposing the following condition:

(9)

8

~.R,=O,

(13)

2q,

(12)

As we shall propose in the next section a suitable set of parameters can be chosen (although there is not a unique possible choice). Here we want to justify the method proposed on the basis of the data on reaction (l) existing from 4 GeV/c to 24.8 GeV/cV). T w o main longitudinal configurations are available in reaction (1) due to the symmetry o f the initial state: pp~p(p~n),

and

pp ~(pvz)(pvz).

SCALING METHOD FOR SUPER-HIGH ENERGIES

o(mb)

I

Tl'

• (7 p (pTrrr)

241 I I

i

16-

+

+ + + 08

557

The corresponding cross sections as a function of laboratory beam momentum are shown in fig. 2. Their tendency to flatten off at high energies is clearly seen; this supports, at least partially, the second "basic principle" assumed; i.e. that present accelerator energies are not far from the asymptotic region. Fig. 3 shows to what extent a scaling hypothesis may hold. In this figure one can compare the distributions of the "reduced longitudinal m o m e n t a " x I defined by eq. (6) at different energies; the differences in these distributions tend to vanish with increasing energy. The main question is whether or not the population of final particles having x, ~ 0 vanishes for s--* oo for an exclusive channel such as reaction (l).

÷ ÷

oi

10

20

24 8 GeV/c

30 PLAII(GeVlc)

21 8

GeVlc//"

19 GeV/c/

Fig. 2. Total cross-sections (4 prongs) for the reactions pp~p(p2zt) and pp~(p~)(pzt) vs incident momentum.

10 GeV/c 8 GeV/c , 6

GeV/c l

4 95 GeV/c t 4 GeV/c%¢ ]

/

11

20

/

/ L

10"

-1

~-'-

0

+1

Xp

t

0

4

2

6

8

1

i x ~1 (a)

Fig. 3.(a) \

) " 100 distributions for "real" plons at

NTOT -/

same energies of ref. 7; (b) \ NTOT // 100 distributions for "real" protons at same energies of ref. 7.

Transverse momentum distributions are not sensitive to s. For sake of simplicity we shall call " M a x i m u m Pionization Method" (MPM) the method obtained by using the assumptions (12) and (13) while we shall call "Pure Fragmentation Method" (PFM) that obtained by using the assumptions (15), (16) and (17').

200 GeV/c 150 G 100 G

e

V

e

V

~

~

65 GeVlc %

24 a )

... 16

J

i ~

loo

I

~-

.~

I rtn 2o0

201

4

.-

4

6

L~

8

-

.

11

,8:D

'

2

..~-

"[

11~J j l

.

"

~I

~

/

P ~- ( GeV/c )

Frog. 4. \ NTOT ./ 100 d~smbutlons obtained by MPM at 65, 100, 150, 200 OeV/c mczdent momentum for: (a) generated last protons, (b) generated slow protons, (c) generated ~ -, (d) generated ~-.

200 GeV/c

]

lOO G~V/c~ ~

1[

6s G ~ v / ~ ~ ' ' ' ~ j

""

1

i

a)

b)

-1

c)

-1

0

56

co= op '

56 'i. ~ . ~ "r ,_r J ~ - ~ _ J . -== ~

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-59

j:

co, o . +

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IL~

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2

. . . . >~,6o ~56'

./:_ __.._,,..~

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d)

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0

+1

Cos Oar-

(dN/dO "~. 100 for generated \ NTOT J (dN/dcosO'~ 100for generated slow protons, (c) (dN/dcosO'~. (dN/dcosO'~. 100 fast protons, (b) ~, N~-~o r 7" \ NTOT ,,] lOO for generated n+, (d) Fig. 5. Angular d,str,butions obtained by MPM at 65, 100, 150, 200 GeV/c inc,dent momentum: (a)

~, NTOT /

for generated ~-.

560

G. CECCHET et al.

3. Maximum pionization method (MPM) In this method for each final particle we assume R l - - r~,

~/s2 Q, = ~ q , ,

08)

1

where capital letters refer to high energy quantities, small letters to the existing measured ones. For each event we define an s~ s)

x/s~ = Z~/(Q2,+r~+m2,).

(19)

i

This general assumption will preserve the "fireballs" due to the fact that the "stretching" introduced by the k,'s is acting in the same direction of the actual momenta qi. N being the total number of outgoing particles, n(m) the number of forward (backward) going particles in the cms, excluding the two "leading particles" qv, q,, we have

n+m= N-2. Thus energy conservation leads to:

~ k , = x/s 2 - ~ Iq,I; 1

Furthermore we calculate "p for the Lorentz transformation matrix L('7) in order to obtain the laboratory four-momenta/3 from CMS four-momenta/~,

( ~ Iq,[ ~ x/s1),

while momentum conservation implies: i

tl+m

kv+

k , = k s+ 3=1

~;, =

L- '(,~)~7,.

(20)

We shall now show a set of distributions of variables suitable to explain the characteristic effects of MPM at four different energies, i.e. 65 GeV, 100 GeV, 150 GeV and 200 GeV*. Fig. 4 summarizes the laboratory momentum distributions of each final particle. While the leading particle is increasingly forward peaked, the pions accumulate at low momentum values although they spread up to the maximum allowed momentum. In fig. 5 we display the single particle angular distributions in the laboratory frame. The forward peaking of the leading particle is best seen here. For the pions it is very interesting to point out that a fraction of the order of 10% is emitted backward in the laboratory system at all energies. Quite a different behaviour is shown by the slow protons for which the backward fraction tends to decrease with increasing energy.

Q, = qt+ Iqd q' k "

(kt > = 0).

(21)

* We have used 6584 events only (from 19 to 24.8 GeV/c incident momentum) to simulate these super-high energy events.

Y~ k,.

(23)

l=n+l

A minimum arbitrary assumption could be kv = k l = k 2 . . . . .

k,, (24)

k s = k.+ 1 = kn+2 . . . .

= kn+m.

From eq. (23) momentum conservation is satisfied if m+l kr = ~ ks.

(25)

From eq. (22) energy conservation is satisfied if nh-ffl

~, k , + k v + k s = , / s 2 -

~.lq,I,

(26)

l

l=l

that is, introducing eq. (25) into eq. (26): ~/s2- Z Iq,I t

kr-

2(n+l) ' x/s2- ~ Iqil

(27)

1

ks = 4. Pure fragmentation method (PFM) [n this method we have to introduce some free parameters k,, the maximum number of which is equal to the number N of outgoing particles. [n order to freeze the "stretching factor" [q,,.x-q~[, where [q=~xl=½x/st is the maximum longitudinal momentum available, we assume for each final particle:

(22)

1

2(re+l)

If assumptions different from eq. (24) are adopted, the equivalent algebra has to be explicitely developed. As an example, in applying PFM to reaction (1), we assume a. when the two pions are emitted in the same cms direction kF = ks = ½(~/s2- ~ [q,I), ' k+ = k_ = 0 ,

(27')

where obviously k+ (k_) refers to the positive (negative) pion;

200 GeWc 150 GeV/c 100 GeV/c 65 GeV/c

261 a)

8

100

2OO

12

8

b)

18 1

24

2 I

Ps C~,,/e

19

2)

L

12

e) 8

2

4

6

|

19 17

21 19 16

8

P,,+(~v/O 15

8

P~r- (~vld

b

d) 12

2 Fig. 6. ~

4

6

" 100 distributions obtained by PFM at 65, 100, ! 50, 201) GeV/e incident momentum for: (a) generated fast protons, (b) generated slow protons, (c) generated ~+, (d) generated ~-.

200 GeV/c /

150 G e V / c / 100 GeV/c~ / jl --/

65 6ev/c _.Jj Z ~

]~

"¢

i

,

tL

a)

/ / " L

b-~_,

1o

20

.r~

eD(mrad) "

,_-
; Jl

r~"

W--!.

--S -

"

~--"

_,!_>.¢_--:-, . J-

--

-59

0

~ ~ '

° ~~ ' . . . .

o

d)

0

I

-

/J"

M

~

"~-~o

L °°

~ "

r

~~

~

Z

.-J 166

i" ?r

L -

'

Ji ~

(-~o

~

r-.--"--'

-S-

.

"

? %J

[--"

/L

-

~s9 k

-1

[.=

_-~-

_ J-_ "

-,

[.s6

~]

~

'

r

~ ...... ~

~

-

r-=, (]6

_~Jb

2

t/-/

+1

Cos O~.Fig. 7. Angular distributions obtained by PFM at 65, 100, 150, 200 GeV/c inmdent momentum' (a) \ NTOT /] 100 for generated

(dN/dcosO% (c,(dN/dc°sO~ • 100 for generated ~ +, (.d)(dN/dc°sO~ 100 NT~T ")" 100 for generated slow protons, \ N'rOT ../ \ NT~-T ".,]"

last protons, (b) \

for generated ~-.

SCALING MLTHOD FOR SUPER-HIGH ENERGIES b. if the two pions are emitted in o p p o s i t e cms directions kF

----

k, = k+ = k_ = at(x/s2- ~ Iq,I).

(27")

1

W e shall now show for P F M using eqs. (27') a n d (27"), the same distributions which we have a l r e a d y discussed for M P M * . The m o m e n t u m distributions o f the final particles are shown in fig. 6. F o r the leading p r o t o n r o u g h l y 36% o f the events have nearly the m a x i m u m allowed m o m e n t u m at every energy; they clearly belong to the kinematical configuration o f the diffractive type; the remaining events are evenly distributed, reaching d o w n to the low m o m e n t u m region. The tendency o f the pions to be p r o d u c e d at low m o m e n t u m values is also conserved, but the higher m o m e n t u m tail is m o r e densely p o p u l a t e d than for the M P M case. The slight differences observed in the real d a t a between n + a n d ~z- (not shown and due to the A ++ * Also in this case we have considered the same real data used m MPM generation.

563

resonance) is magnified at increasing energies as can be seen by c o m p a r i n g the equivalent distributions for the pions o f o p p o s i t e sign. T u r n i n g now to the a n g u l a r distributions shown in fig. 7 we observe httle difference between P F M a n d M P M . T h e main differences are: a. the higher frequency o f b a c k w a r d particles (shown in table 1)

TABLE

l

PFM percentage of backward particles m the laboratory frame at 65, 100, 150, 200 GeV/c incident momentum Plab (GeV/c)

65

100

150

200

Slowproton ~+ =-

3.2 95 5.0

3.0 95 54

0.8 9.6 5.7

0.5 9.5 0.5

b. the m o r e drastic energy d e p e n d e n c e o f for the slow p r o t o n s (fig. 7b).

da/dcosO

100 GeV/c

50C' P "n'* 30C

~

100! 50"

10(3 3E

¢~ "~

~

15

20

25

~

150.

04

08

12

16

14

18

22

26

20

.

P Tt-

180t

100

12o1

1 i I

Uu

50.

I

10

15

20

25 M (GeV)

Fig. 8. Mass dlstnbuuons for events scaled by MPM at 150 GeV/c incident momentum.

30

564

G. CECCHET ct al.

5. Conclusions The scaling methods presented here conserve most of the dynamical content of the real events. In fig. 8 some resonant masses are plotted. The strong d + + production is preserved and is it indeed remarkable that the shape of the underlying background is not stretched by scaling. The presence of a dip in the distribution of dN/dO for the fast proton at all energies is a direct consequence of the finite mean value of the transverse momenta Pt (,.~ 300 MeV/c), related to the longitudinal momenta by:

Pi ~ Pb,~mO, giving then a peak in the angular distribution. As observed before PFM stretches, even more so than MPM, the center of mass longitudinal quantities, giving rise to a "centrifugal" effect of sorts in the laboratory variables. In conclusion we can completely determine the kinematics of possible super-high-energy processes in the extreme hypotheses of maximum pionization and limiting fragmentation. If the scaling law on which this work is based maintains its validity for higher multiplicities and different reactions, then it is possible to extrapolate these procedures to them. The computing time spent to generate events on the UNIVAC 1106 computer at the University of Milan is about 6 x l0 -2 s/event and this is the substantial advantage of this method over the usual Monte Carlo procedures. We thank all experimenters and laboratories who have made their data available to the world collaboration. The effort done by E. U. Colton of LBL in collecting and combining all the Data Summary Tape into a unique set is gratefully acknowledged.

Tovey, Compilation of cross-sect~ons, II - Antlprotons Induced reactions, CERN HERA 70-3 (Aug. 1970). d - E . Flamtmo, J. D. l-Jansen, D. R. O. M o m s o n and N. Tovey, Compdations of cross-sections, I I I - K + reduced reactions, CERN HERA 70-4 (Sept. 1970). e - E Flaminlo, J. D. Hanson, D. R. O M o m s o n and N Tovey, Compilation of cross-sections, IV - ~ ¢ induced reactions, CERN HERA 70-5 (Sept. 1970). f - E. Flamlmo, J. D. Hanson, D. R. O. Morrison and N. Tovey, Compilation of cross-sections, V - K - reduced reactions, CERN HERA 70-6 (Oct. 1970). g - E. Flaminio, J. D. Hansen, D. R O. Morrlson and N. Tovey, Compilation of cross-sections, VI - ~ - induced reactions, CERN HERA 70-7 (Oct. 1970). ~) S. Rattl, Multiparticle high energy reactions, Lectures to the 1971 School Computer in science, Report I F U M 126 HE (ICTP, Trieste, 1971). a) M. Koshlba, Progr. Theoret. Phys. (Kyoto) 37 0967) 1942; Can. J. Phys. 46, Supplement (1968). 4) K. Schlupmann, Particle production experiments, Lecture Notes VARENNA 0971). 5) j. Benecke, T. T. Chou, C. N. Yang and E. Yen, Phys. Rev. 1 8 8 (1969) 2159; R. P. Feynman, Phys Rev. 23 0969) 1415, H. Cheng and T. T. Chou, Phys. Rev. Letters 23 0969) 1311. 6) A. W. Kittel, S. Ratti and L. Van Hove, Nucl. Phys B 30 (1971) 333. 7) World Data Summary Tape on the reaction p p - ~ - p p ~ from 4.0 GeV/c to 24.8 GeV/c. Laboratories involved: 4.0 GeV/c Milano University, Genova University 4.95 GeV/c Princeton University 6.0 GeV/c Lawrence Radmtion Laboratory, SLAC 6.0 GeV/c Genova University, Milano Umversity and Oxford University 6.6 GeV/c University of Cahforma (UCRL) 6.92 GeV/c Weizmann Institute 8.0 GeV/c Illinois Umverslty 10.0 GeV/c Cambridge Institute and Hamburg-Desy 19.0 GeV/c Scandinavian Collaboration 21.8 GeV/c Iowa University 24.8 GeV/c Columbia University. 8) s2* is different from s2 the more the event is transversal. In order to avoid this point we have also adopted the approximate formula:

Q,-'

References i) a - P. Spillantim and V. Valente, A collection of plon photoproduction data, CERN HERA 70-1 (June 1970). b - J. D. Hansen, D. R. O. Momson, N Tovey and E. Flammlo, Compilation of cross-sections, I - Proton reduced reactions, CERN HERA 70-2 (July 1970). c - E. Flammio, J. D. Hansen, D. R. O. Morrison and N.

q,,

~lq,I I

where

l

t