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Long-range correlations as a phenomenological probe of non-scaling behaviour
Volume 57B, number 4
PHYSICS LETI'ERS
21 July 1975
L O N G ~ R A N G E C O R R E L A T I O N S AS A P H E N O M E N O L O G I C A L PROBE OF NON-SCALING BEHAVIOUR J. KWIECINSKI* and R.G. ROBERTS
Theory Division,RutherfordLaboratory, Chilton,Didcot Oxen 0Xll OQX, UK Received 23 May 1975 Since pure scaling terms give no correlations in the double fragmentation region, then the magnitude of such a correlation provides a Sensitive measure of the non-scaling terms. The normalised correlation R for pp ~ pp + X at one ISR energy will allow one to distinguish the various tripleRegge analyses of pp ~ pX. The correlations at lower energies for non-diffractive processes are also discussed. Our understanding of single particle inclusive distributions for the fragmentation process b a c has advanced enormously by describing such a reaction in the Mueller-Regge framework. The factorisation properties of both Pomeron and normal Regge exchanges leads to many relations which are successfully satisfied by the experimental data [e.g. 1]. In these processes, the Pomeron exchange governs that part of the cross-section which scales while the Regge exchange terms control the non-scaling contribution. For non-exotic processes the separation of these two contributions involves studying the energy dependence of the cross-section at a fixed x or M2/s and extracting the constant component from that which falls like s -1/2. This is not always so straightforward in practise as it sounds. The error bars are usually quite large and in order to get as large a variation in s -1/2 as possible one has to use low energy data. This means that the fragmentation region may be dominated by resonances and then finite-mass-sum rules have to be evaluated to study the energy.variation [2]. These problems are particularly relevant to the case of diffractive inclusive reactions where there is especial interest in determining the scaling and non-scaling contributions (PPP and PPM respectively, in the triple-Regge language) since each of these leads to distinctive properties for the inelastic diffractive cross-section. Several triple-Regge analyses have been carried out on pp -" pX [3,4] which, because of the above practical ambiguities, lead to differing estimates of the relative magnitudes of the non-scaling and scaling terms. This in turn leads to considerable variation in the estimates for the rise of the inelastic cross-section with energy. In this note we propose how, using factorisation, the measurement of the long-range correlations in the double fragmentation region can be used as a sensitive probe for the ratio of the non-scaling to scaling terms. The fact that such a correlation falls with energy like s -1/2 is compensated by the "singular" behaviour in x 1, x 2 and implies that even at ISR energies the correlation may be quite large indeed. Consider the double-fragmentation process a + b --' c + d + X shown in fig. 1 (a). The two-particle distribution is given by
p(2)(tl, M2/s; t2, M2/s; s) = ~k ~kC(tl, M2/s) pbd(t2, M2/s) (M2) ak(O)-1, where k = P, p - f - w - A
(1 a)
2 etc.
and the single-fragmentation process a + b ~ c + X is given by fig. 1 Co) p(1)(t 1, M2/s; s) =
~k ~ ( t l , M2/$)[jbb(o) (M 2) ak(O- 1,
(lb)
* On leave from the Institute of Nuclear Physics, Cracow. 349
Volume 57B, number 4
PHYSICS LETTERS •
O
C
C
> O
i
21 July 1975 ~
(
~
O • ~
•
C
I
O
>
b
b
b d
•
•
>
tb)
d
(a)
Fig. 1. (a) MueUer graph for the double-fragmentation reaction ab ~ cd + X. (b) Mueller graph for the fragmentation reaction ab~c+X. and similarly for a + b -* d + X. Using factorisation one can predict p(2) having extracted the vertices pac from single particle data [5]. Here we shall use factorisation to obtain the normalised correlation defined as* R =
ab T p(2)(t I , M21s., t 2, M2/s,s) oTO P(1)(t 1 , M21/s; s) p(1)(t 2,
M2/s; s)
1.
(2)
In the high-energy limit we have
R
s - 1 / 2 ~M ti, acrt 1, M2/s 2 bd • M2/s'~[jbb[ [ Mk lC !~ l-,bdQ, M t M2/s~abba~(M2/s~-l/2 21 )vp ~'p ~ r J - I ~ pae( t l , M1/S)FM(t2 21) P ja~(M2/s'~-ll2 M~ 2 / ) 2 a~fJ~d bb (M1/s) 2 -1/2 - Fpbd(t2, M 22/ s ) F ~ac( t l , M1/s)fJP
(3)
+ where the summation extends over normal Regge trajectories (i.e. not Pomeron). At higii energies we have
M2/s and so
R ~ , s -112 ~JF~FbMd(1--1Xll)-ll2(1--1x21)-ll2--{3bf~[J# --FpaC i~bd ~pbb/3~ ) (a1 _ ~ix2 i -1/2 M b'b a~//o(1) (tl, M21/s)o(1)(t2, M2/s). - F p bd F ~a cl p, BbG ~ ( 1 - 1 X l l )-1/2 + P pac Fpbd /7~13~
(5)
I--
From (5) we see that even at high energies, if we go close to Ix I l, Ix21 ~ 1 then the value of R will not necessarily be small. At high energies we have, roughly g ~
zMr r d (1 r cr d
Ix11)-1/2 (1 -Ix21)-l/2s -1/2,
(6)
and so the magnitude of R provides, in principle, a sensitive measure of the ratio of non-scaling/scaling terms. If the fragmentation cross-section was purely scaling then the correlation would be identically zero, provided, of course, that the scaling piece factorizes. Thus a careful measurement of the correlation at one energy is an alterna* It is very important that we normalise R using OTOT not OlNEL. 350
Volume 57B, number 4
PHYSICS LETTERS
21 July 1975
1.0
i ,
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0.9
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0,08
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R (X~, Xi ) -X~ = Xz
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~ ~6 ~
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Fig. 2. Normalised correlation R (x t, x 2 ) along the diagonal where x = x 1 = - x 2. The curves correspond to predictions based on Triple-Regge analyses of refs. [3,4, 6].
Fig. 3. Normalised correlation R (x 1, x2) in the double fragmentation region of K-p ~ ~. Op + X at 10 GeV/c. Results computed from the data of ref. [8]. The contours are the prediction of a naive FDP model described in the text.
tive to measuring the single particle distributions over many energies, as far as determining the magnitude of the non-scaling piece. To see in practise how sensitive a test this is we study the example mentioned earlier, pp -~ pp + X. TripleRegge analyses differ significantly on the ratio PPM/PPP as determined from a study of the energy dependence of pp -~ pX. From (6) we see that the magnitude of R for IXll ~ Ix21 ~ 1 is a precise measure of (PPM/PPP) 2. One analysis [3] indicates that the inclusive reaction pp ~ pX almost scales. A consequence of this dominance by PPP is that the inelastic diffraction cross-section increases by 2.3 mb from Plab = 300 ~ 1500 GeV/c. Another analysis [4] which argues for a much stronger PPM at the expense of the PPP predicts a corresponding increase of only 0.44 mb. Therefore it is quite crucial in our understanding of inelastic diffraction to know this PPM/PPP ratio. Fig. 2 shows the predictions of these two analyses based on eq. (5) at an ISR energy of s = 940. The value of R along the diagonal is shown and we see the expected discrepancy in the magnitude of R. Also shown is the prediction according to the so-called "f-dominated Pomeron" (FDP) model [6]. In this model the Pomeron couples to particles and to the fragmentation "blob" via the f-meson (or f'). Consequently the f/P ratio as determined from total cross-sections applies equally well for the fragmentation vertices, i.e.
rfo, M2/s) ~f Fp(t, M 2/s )
-
,6p
= c.
(7)
From analyses of oTOT we get e ~ 1 which is consistent with value obtained for the fragmentation vertices from inclusive data [6]. In this case of pp -> pX, we expect the meson exchange term (at least in the triple-Regge limit) to be entirely f exchange. Hence from eq. (5) we get, as s ~ o o
R ~ c2s -1/2 {(1-Ix11)-1/2
(1 - I x 2 l ) -1/2
- (1
-Ixll )-1/2 _
(1 -Ix21) -1/2 + 1}.
(8)
This prediction is also included on fig. 2. We see that, in practise, the measurement of the correlation between 351
Volume 57B, number 4
PHYSICS LETTERS
21 July 1975
two fast protons at the ISR will serve as a sensitive test of the PPM/PPP ratio - a quantity still apparently not yet pinned down from single proton spectra. Recent data from NAL suggests indirectly a very large PPM term, to the extent that the diffractive cross-section remains constant from 100 to 400 GeV/c [12]. The implication of this is that we should see a correspondingly huge correlation between the produced protons. Such an experiment is not easy to perform but there is a strong likelihood of pp ~ pp + X being measured in the near future at ISR. The interest of such an experiment lies mainly in the expectation of observing double-Pomeron exchange, the crosssection for which can be estimated by factorisation from the single particle diffractive cross-sections. We therefore see that added important information can be learnt in this exciting experiment. Next we ask whether there is already any evidence for positive longitudinal correlations in the double fragmentation region. One problem is that interest in correlations has concentrated on the central region rather than on fragmentation regions. Consequently the plots were made in rapidity y, or 7/= In (tan0/2) rather than x. If we turn to the ISR data of Allaby et al. [7] we see correlations between a lr- at fixed positive x (0.8 for example) and a charged particle in the opposite hemisphere, whose angle is measured. Nevertheless R has a sharp positive spike for 7/~ - 5 which is precisely the type of correlation we expect if there is a significant non-scaling term. In this reaction the charged particle producing the strong positive correlation must be a nucleon. A pion, or kaon, in the backward hemisphere would produc e no correlation with the forward ~r- since then the ab~a system would be exotic and only Pomeron exchange would be allowed. We cannot quantitatively compare the magnitude of this spike with models since only ,/is measured and R was normalised by OlNEL. However we would like to point out that this correlation is a dynamic effect, not purely kinematic as suggested in ref. [7] since if the reaction was scaling with energy no such correlation would be seen. If we compare pp - ' 7r-lr+X and n-p -, lr-lr+X at 205 GeV[c, the correlations in the first case are zero]or negative) for large positive Y~r- and large negative yn, while there is a small positive enhancement in the same region for the second case. While we again cannot quantitatively analyse the data for the same reasons as above, this qualitative behaviour is what one would expect. The positive long-range correlation does not depend on assuming triple-Regge behaviour. Only factorisation of the fragmentation vertices r ac (t, M2/$) is assumed and so we can expect to see such effects at comparatively low energies. The ABCLV collaboration on K-p at 10 GeV/c have analysed K-p --, ~Op + X in the double fragmentation region [8]. From the published values of p(2) and p(1) we have computed the values o f R (normalising to oKol~ ) . In fig. 3 we show the results versus X~,o and Xp. The errors are large since we have to compute the differeri6e-of quantities each with typically 20% erro-rs. Nevertheless the trend is what one would expect from eq. (5) namely a peaking towards IXll ~, Ix21 ~ 1 and roughly equal values along the lines x I - x 2 = const. Notice that the correlation can become very large in this region. In order to make some comparison with a model we have computed the values of an FDP type of model [6]. In general for this reaction we expect, P, f, co, p, A 2 all to be exchanged in fig. 1 (a), (b). However if we were to go to triple-Regge region only P, f would be allowed since the other exchanges would correspond to interference between the exchanges in the t ~ 0 channel which is not allowed by EXD. If we extend this constraint to the whole fragmentation region and use FDP we get the subasymptotic form of eq. (8) which corresponds to the contours also plotted in fig. 3. We see that the model describes the general trend but not the actual magnitude of R. The easiest conclusion is that the constraint of only P and f exchange is too strong and that the other Regge exchanges should be included which would greatly increase the magnitude ofR. Since the double fragmentation correlation is a sensitive measure of the non-scaling contribution it is therefore a convenient way of studying anomalous non-scaling terms. In single fragmentation processes which correspond to reggeon K* - K scattering (i.e. p K__~A)there can be ¢ - f' exchange in addition to the normal ~ = ~ exchanges. This contribution would be dual to the resonances p, f, g, 7/etc. in the M 2 channel while the p - f - c o - A 2 exchange would be dual to the direct resonances 4, f'- Inami and Miettinen made a detail study of such fragmentation processes and found the f - ~ ' exchange, characterised by a s -1 fall off, to be surprisingly large [9]. By similar arguments to those above we could hope to study this contribution by considering the correlations in K-p ~ Ir-A + X or ~r-p --, K°A + X for example since we would then have reggeon K*-reggeon I( ° scattering. We 352
Volume 57B, number 4
PHYSICS LETTERS
21 July 1975
would therefore suggest that such processes be measured experimentally to see if the correlations are indeed large. We can, in fact, make a crude estimate on the basis o f factorisation for the magnitude o f R. Even though the energy dependence is now much steeper, the correlation R is more "singular" i.e. ~ (1 - [Xll) -1 (1 - I x 2 1 ) -1 . I f we take Ff(t, M2/s)/rp(t, M2/s) = c ~ 1 from FDP fits and Ff,(t, M2/s)/rp(t, M2/s~ = d ~ l0 from fitting data in ref. [9] and invoking that analysis to neglect the p - f - ~ o - A 2 exchanges in " K K " scattering we get for K-p-* n-A + X
2) (1 - I X l l ) + 2c2s-l/2(1-[Xll) -1/2] [1 +d2s-l(1-]x2]) -1]
S-1 d 2 [(1 - [ X l l ) -1 (1 - I x 2 1 ) -1 - (1 - x 2 ) - 1 - 2 ( c 2 / d R"
[I
-1/2 ]
'
(9)
we find at s = 30 that R (x 1 = 0.6, x 2 = - 0.6) ~ 0.5 rising sharply to R (x I = 0.75, x 2 = - 0.75) = 1.4. It will be very interesting to see if such very large double fragmentation correlations actually are seen experimentally. In conclusion we see that measurements o f double fragmentation correlations at a singl~ energy can lead to accurate evaluations o f the non-scaling terms which could only otherwise be got from a study o f single fragmentation cross-sections over a wide range o f energies. Furthermore we see that in many cases a large correlation is expected on the basis o f factorisation. We are grateful to Ryszard Stroynowsld for informing o f the cross-section for K - p -* ~ ° p X . One o f us (JK) thanks R.J.N. Phillips for hospitality at Rutherford Laboratory.
References [ 1 ] H.I. Miettinen, Regge phenomenology of inclusive reactions, thesis 1973; R.G. Roberts, Scottish Summer School Lectures 1973. [2] H.M. Chart, H. Miettinen and R.G. Roberts, Nucl. Phys. B54 (1973) 411. [3] D.P. Roy and R.G. Roberts, Nucl. Phys. B77 (1974) 240. [4] R.D. Field and G.C. Fox, Nucl. Phys. B80 (1974) 367. [5] R.C. Brewer, J. Ellis and R.N. Cahn, Phys. Lett. 44B (1973) 81. [6] T. Inami and R.G. Robert, Rutherford preprint - RL-75-025 to be published in Nucl. Phys. [7] J.V. Allaby et al., Correlations between charged particles and one momentum-analysed forward negative particle at the ISR, paper submitted to London ConL, July 1974. [8] J.V. Beanpre et al., Aachen-Bonn-CERN-London-Vierma collaboration, Phys. Lett. 40B (1972) 510. [9] T, Inami and H.I. Miettinen, Phys. Lett. 49B 0974) 67. [ 10] J.V. Beanpre et al., Aachen-Berlin-CERN-London-Vienna and Bruxelles - CERN Collaborations, Nucl. Phys. B30 (1971) 381. [11 ] Bosetti et al., Aachen-Berlin-CERN-London (i.C.)-Vienna Collaboration, Nucl. Phys. B60 (1973) 307. [12] R.D. Schamberger et al., Phys. Rev. 34 (1975) 1121.
353
PHYSICS LETI'ERS
21 July 1975
L O N G ~ R A N G E C O R R E L A T I O N S AS A P H E N O M E N O L O G I C A L PROBE OF NON-SCALING BEHAVIOUR J. KWIECINSKI* and R.G. ROBERTS
Theory Division,RutherfordLaboratory, Chilton,Didcot Oxen 0Xll OQX, UK Received 23 May 1975 Since pure scaling terms give no correlations in the double fragmentation region, then the magnitude of such a correlation provides a Sensitive measure of the non-scaling terms. The normalised correlation R for pp ~ pp + X at one ISR energy will allow one to distinguish the various tripleRegge analyses of pp ~ pX. The correlations at lower energies for non-diffractive processes are also discussed. Our understanding of single particle inclusive distributions for the fragmentation process b a c has advanced enormously by describing such a reaction in the Mueller-Regge framework. The factorisation properties of both Pomeron and normal Regge exchanges leads to many relations which are successfully satisfied by the experimental data [e.g. 1]. In these processes, the Pomeron exchange governs that part of the cross-section which scales while the Regge exchange terms control the non-scaling contribution. For non-exotic processes the separation of these two contributions involves studying the energy dependence of the cross-section at a fixed x or M2/s and extracting the constant component from that which falls like s -1/2. This is not always so straightforward in practise as it sounds. The error bars are usually quite large and in order to get as large a variation in s -1/2 as possible one has to use low energy data. This means that the fragmentation region may be dominated by resonances and then finite-mass-sum rules have to be evaluated to study the energy.variation [2]. These problems are particularly relevant to the case of diffractive inclusive reactions where there is especial interest in determining the scaling and non-scaling contributions (PPP and PPM respectively, in the triple-Regge language) since each of these leads to distinctive properties for the inelastic diffractive cross-section. Several triple-Regge analyses have been carried out on pp -" pX [3,4] which, because of the above practical ambiguities, lead to differing estimates of the relative magnitudes of the non-scaling and scaling terms. This in turn leads to considerable variation in the estimates for the rise of the inelastic cross-section with energy. In this note we propose how, using factorisation, the measurement of the long-range correlations in the double fragmentation region can be used as a sensitive probe for the ratio of the non-scaling to scaling terms. The fact that such a correlation falls with energy like s -1/2 is compensated by the "singular" behaviour in x 1, x 2 and implies that even at ISR energies the correlation may be quite large indeed. Consider the double-fragmentation process a + b --' c + d + X shown in fig. 1 (a). The two-particle distribution is given by
p(2)(tl, M2/s; t2, M2/s; s) = ~k ~kC(tl, M2/s) pbd(t2, M2/s) (M2) ak(O)-1, where k = P, p - f - w - A
(1 a)
2 etc.
and the single-fragmentation process a + b ~ c + X is given by fig. 1 Co) p(1)(t 1, M2/s; s) =
~k ~ ( t l , M2/$)[jbb(o) (M 2) ak(O- 1,
(lb)
* On leave from the Institute of Nuclear Physics, Cracow. 349
Volume 57B, number 4
PHYSICS LETTERS •
O
C
C
> O
i
21 July 1975 ~
(
~
O • ~
•
C
I
O
>
b
b
b d
•
•
>
tb)
d
(a)
Fig. 1. (a) MueUer graph for the double-fragmentation reaction ab ~ cd + X. (b) Mueller graph for the fragmentation reaction ab~c+X. and similarly for a + b -* d + X. Using factorisation one can predict p(2) having extracted the vertices pac from single particle data [5]. Here we shall use factorisation to obtain the normalised correlation defined as* R =
ab T p(2)(t I , M21s., t 2, M2/s,s) oTO P(1)(t 1 , M21/s; s) p(1)(t 2,
M2/s; s)
1.
(2)
In the high-energy limit we have
R
s - 1 / 2 ~M ti, acrt 1, M2/s 2 bd • M2/s'~[jbb[ [ Mk lC !~ l-,bdQ, M t M2/s~abba~(M2/s~-l/2 21 )vp ~'p ~ r J - I ~ pae( t l , M1/S)FM(t2 21) P ja~(M2/s'~-ll2 M~ 2 / ) 2 a~fJ~d bb (M1/s) 2 -1/2 - Fpbd(t2, M 22/ s ) F ~ac( t l , M1/s)fJP
(3)
+ where the summation extends over normal Regge trajectories (i.e. not Pomeron). At higii energies we have
M2/s and so
R ~ , s -112 ~JF~FbMd(1--1Xll)-ll2(1--1x21)-ll2--{3bf~[J# --FpaC i~bd ~pbb/3~ ) (a1 _ ~ix2 i -1/2 M b'b a~//o(1) (tl, M21/s)o(1)(t2, M2/s). - F p bd F ~a cl p, BbG ~ ( 1 - 1 X l l )-1/2 + P pac Fpbd /7~13~
(5)
I--
From (5) we see that even at high energies, if we go close to Ix I l, Ix21 ~ 1 then the value of R will not necessarily be small. At high energies we have, roughly g ~
zMr r d (1 r cr d
Ix11)-1/2 (1 -Ix21)-l/2s -1/2,
(6)
and so the magnitude of R provides, in principle, a sensitive measure of the ratio of non-scaling/scaling terms. If the fragmentation cross-section was purely scaling then the correlation would be identically zero, provided, of course, that the scaling piece factorizes. Thus a careful measurement of the correlation at one energy is an alterna* It is very important that we normalise R using OTOT not OlNEL. 350
Volume 57B, number 4
PHYSICS LETTERS
21 July 1975
1.0
i ,
......
0.9
f -
0,08
0.8
/
0.7 0.6
R (X~, Xi ) -X~ = Xz
0.5
. . . . . . . .
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F'~d + Fox i 5o~n. 1 /
/
OA
0.2 I t
/ 0.3
',/o22 .
.
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.
I
0.25 Ii - ~-o.6
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/o.z,
'/
/I', ._o.~9 4 .
- 50 ~
l/0.56 /I" ~0.62
/I
q
/
J /
0.20 *-0.51
z0./d
0.2
- / - -,- - ; / -
V
~ ~6 ~
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na'J
,-ti7o/;7 "'*~
t / I
1.22/ 1
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~/_~--I
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-" a- -- - _ '--"--'-I. . . . . .
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0.7
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a- . . . . .
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Fig. 2. Normalised correlation R (x t, x 2 ) along the diagonal where x = x 1 = - x 2. The curves correspond to predictions based on Triple-Regge analyses of refs. [3,4, 6].
Fig. 3. Normalised correlation R (x 1, x2) in the double fragmentation region of K-p ~ ~. Op + X at 10 GeV/c. Results computed from the data of ref. [8]. The contours are the prediction of a naive FDP model described in the text.
tive to measuring the single particle distributions over many energies, as far as determining the magnitude of the non-scaling piece. To see in practise how sensitive a test this is we study the example mentioned earlier, pp -~ pp + X. TripleRegge analyses differ significantly on the ratio PPM/PPP as determined from a study of the energy dependence of pp -~ pX. From (6) we see that the magnitude of R for IXll ~ Ix21 ~ 1 is a precise measure of (PPM/PPP) 2. One analysis [3] indicates that the inclusive reaction pp ~ pX almost scales. A consequence of this dominance by PPP is that the inelastic diffraction cross-section increases by 2.3 mb from Plab = 300 ~ 1500 GeV/c. Another analysis [4] which argues for a much stronger PPM at the expense of the PPP predicts a corresponding increase of only 0.44 mb. Therefore it is quite crucial in our understanding of inelastic diffraction to know this PPM/PPP ratio. Fig. 2 shows the predictions of these two analyses based on eq. (5) at an ISR energy of s = 940. The value of R along the diagonal is shown and we see the expected discrepancy in the magnitude of R. Also shown is the prediction according to the so-called "f-dominated Pomeron" (FDP) model [6]. In this model the Pomeron couples to particles and to the fragmentation "blob" via the f-meson (or f'). Consequently the f/P ratio as determined from total cross-sections applies equally well for the fragmentation vertices, i.e.
rfo, M2/s) ~f Fp(t, M 2/s )
-
,6p
= c.
(7)
From analyses of oTOT we get e ~ 1 which is consistent with value obtained for the fragmentation vertices from inclusive data [6]. In this case of pp -> pX, we expect the meson exchange term (at least in the triple-Regge limit) to be entirely f exchange. Hence from eq. (5) we get, as s ~ o o
R ~ c2s -1/2 {(1-Ix11)-1/2
(1 - I x 2 l ) -1/2
- (1
-Ixll )-1/2 _
(1 -Ix21) -1/2 + 1}.
(8)
This prediction is also included on fig. 2. We see that, in practise, the measurement of the correlation between 351
Volume 57B, number 4
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two fast protons at the ISR will serve as a sensitive test of the PPM/PPP ratio - a quantity still apparently not yet pinned down from single proton spectra. Recent data from NAL suggests indirectly a very large PPM term, to the extent that the diffractive cross-section remains constant from 100 to 400 GeV/c [12]. The implication of this is that we should see a correspondingly huge correlation between the produced protons. Such an experiment is not easy to perform but there is a strong likelihood of pp ~ pp + X being measured in the near future at ISR. The interest of such an experiment lies mainly in the expectation of observing double-Pomeron exchange, the crosssection for which can be estimated by factorisation from the single particle diffractive cross-sections. We therefore see that added important information can be learnt in this exciting experiment. Next we ask whether there is already any evidence for positive longitudinal correlations in the double fragmentation region. One problem is that interest in correlations has concentrated on the central region rather than on fragmentation regions. Consequently the plots were made in rapidity y, or 7/= In (tan0/2) rather than x. If we turn to the ISR data of Allaby et al. [7] we see correlations between a lr- at fixed positive x (0.8 for example) and a charged particle in the opposite hemisphere, whose angle is measured. Nevertheless R has a sharp positive spike for 7/~ - 5 which is precisely the type of correlation we expect if there is a significant non-scaling term. In this reaction the charged particle producing the strong positive correlation must be a nucleon. A pion, or kaon, in the backward hemisphere would produc e no correlation with the forward ~r- since then the ab~a system would be exotic and only Pomeron exchange would be allowed. We cannot quantitatively compare the magnitude of this spike with models since only ,/is measured and R was normalised by OlNEL. However we would like to point out that this correlation is a dynamic effect, not purely kinematic as suggested in ref. [7] since if the reaction was scaling with energy no such correlation would be seen. If we compare pp - ' 7r-lr+X and n-p -, lr-lr+X at 205 GeV[c, the correlations in the first case are zero]or negative) for large positive Y~r- and large negative yn, while there is a small positive enhancement in the same region for the second case. While we again cannot quantitatively analyse the data for the same reasons as above, this qualitative behaviour is what one would expect. The positive long-range correlation does not depend on assuming triple-Regge behaviour. Only factorisation of the fragmentation vertices r ac (t, M2/$) is assumed and so we can expect to see such effects at comparatively low energies. The ABCLV collaboration on K-p at 10 GeV/c have analysed K-p --, ~Op + X in the double fragmentation region [8]. From the published values of p(2) and p(1) we have computed the values o f R (normalising to oKol~ ) . In fig. 3 we show the results versus X~,o and Xp. The errors are large since we have to compute the differeri6e-of quantities each with typically 20% erro-rs. Nevertheless the trend is what one would expect from eq. (5) namely a peaking towards IXll ~, Ix21 ~ 1 and roughly equal values along the lines x I - x 2 = const. Notice that the correlation can become very large in this region. In order to make some comparison with a model we have computed the values of an FDP type of model [6]. In general for this reaction we expect, P, f, co, p, A 2 all to be exchanged in fig. 1 (a), (b). However if we were to go to triple-Regge region only P, f would be allowed since the other exchanges would correspond to interference between the exchanges in the t ~ 0 channel which is not allowed by EXD. If we extend this constraint to the whole fragmentation region and use FDP we get the subasymptotic form of eq. (8) which corresponds to the contours also plotted in fig. 3. We see that the model describes the general trend but not the actual magnitude of R. The easiest conclusion is that the constraint of only P and f exchange is too strong and that the other Regge exchanges should be included which would greatly increase the magnitude ofR. Since the double fragmentation correlation is a sensitive measure of the non-scaling contribution it is therefore a convenient way of studying anomalous non-scaling terms. In single fragmentation processes which correspond to reggeon K* - K scattering (i.e. p K__~A)there can be ¢ - f' exchange in addition to the normal ~ = ~ exchanges. This contribution would be dual to the resonances p, f, g, 7/etc. in the M 2 channel while the p - f - c o - A 2 exchange would be dual to the direct resonances 4, f'- Inami and Miettinen made a detail study of such fragmentation processes and found the f - ~ ' exchange, characterised by a s -1 fall off, to be surprisingly large [9]. By similar arguments to those above we could hope to study this contribution by considering the correlations in K-p ~ Ir-A + X or ~r-p --, K°A + X for example since we would then have reggeon K*-reggeon I( ° scattering. We 352
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21 July 1975
would therefore suggest that such processes be measured experimentally to see if the correlations are indeed large. We can, in fact, make a crude estimate on the basis o f factorisation for the magnitude o f R. Even though the energy dependence is now much steeper, the correlation R is more "singular" i.e. ~ (1 - [Xll) -1 (1 - I x 2 1 ) -1 . I f we take Ff(t, M2/s)/rp(t, M2/s) = c ~ 1 from FDP fits and Ff,(t, M2/s)/rp(t, M2/s~ = d ~ l0 from fitting data in ref. [9] and invoking that analysis to neglect the p - f - ~ o - A 2 exchanges in " K K " scattering we get for K-p-* n-A + X
2) (1 - I X l l ) + 2c2s-l/2(1-[Xll) -1/2] [1 +d2s-l(1-]x2]) -1]
S-1 d 2 [(1 - [ X l l ) -1 (1 - I x 2 1 ) -1 - (1 - x 2 ) - 1 - 2 ( c 2 / d R"
[I
-1/2 ]
'
(9)
we find at s = 30 that R (x 1 = 0.6, x 2 = - 0.6) ~ 0.5 rising sharply to R (x I = 0.75, x 2 = - 0.75) = 1.4. It will be very interesting to see if such very large double fragmentation correlations actually are seen experimentally. In conclusion we see that measurements o f double fragmentation correlations at a singl~ energy can lead to accurate evaluations o f the non-scaling terms which could only otherwise be got from a study o f single fragmentation cross-sections over a wide range o f energies. Furthermore we see that in many cases a large correlation is expected on the basis o f factorisation. We are grateful to Ryszard Stroynowsld for informing o f the cross-section for K - p -* ~ ° p X . One o f us (JK) thanks R.J.N. Phillips for hospitality at Rutherford Laboratory.
References [ 1 ] H.I. Miettinen, Regge phenomenology of inclusive reactions, thesis 1973; R.G. Roberts, Scottish Summer School Lectures 1973. [2] H.M. Chart, H. Miettinen and R.G. Roberts, Nucl. Phys. B54 (1973) 411. [3] D.P. Roy and R.G. Roberts, Nucl. Phys. B77 (1974) 240. [4] R.D. Field and G.C. Fox, Nucl. Phys. B80 (1974) 367. [5] R.C. Brewer, J. Ellis and R.N. Cahn, Phys. Lett. 44B (1973) 81. [6] T. Inami and R.G. Robert, Rutherford preprint - RL-75-025 to be published in Nucl. Phys. [7] J.V. Allaby et al., Correlations between charged particles and one momentum-analysed forward negative particle at the ISR, paper submitted to London ConL, July 1974. [8] J.V. Beanpre et al., Aachen-Bonn-CERN-London-Vierma collaboration, Phys. Lett. 40B (1972) 510. [9] T, Inami and H.I. Miettinen, Phys. Lett. 49B 0974) 67. [ 10] J.V. Beanpre et al., Aachen-Berlin-CERN-London-Vienna and Bruxelles - CERN Collaborations, Nucl. Phys. B30 (1971) 381. [11 ] Bosetti et al., Aachen-Berlin-CERN-London (i.C.)-Vienna Collaboration, Nucl. Phys. B60 (1973) 307. [12] R.D. Schamberger et al., Phys. Rev. 34 (1975) 1121.
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