- Email: [email protected]
c
Nuclear Physics B102 (1976) 4 0 5 - 4 2 8 © North-Holland Publishing Company
A STUDY OF TRANSVERSE AND LONGITUDINAL MOMENTUM CORRELATIONS IN PROTON DIFFRACTION DISSOCIATION AT 14 GeV/c *
W. OCHS *, V. DAVIDSON, A. DZIERBA **, A. FIRESTONE ¢, W. FORD **, R. GOMEZ, F. NAGY, C. PECK and C. ROSENFELD California Institute of Technology, Pasadena, California, 91125 J. B A L L A M , J. C A R R O L L , G. C H A D W I C K , D. L I N G L I N * a n d K. M O F F E I T * *
Stanford Linear Accelerator Center, Stanford, California, 94305
R. ELY, D. GRETHER and P. ODDONE Lawrence Berkeley Laboratory, Berkeley, California, 94 720 Received 6 October 1975 We present data on momentum correlations both in and out of the production plane for the reactions n+p--* n~(n+n) and n - p ~ n - 0 r - A ++) The dissociation products f show strikingly similar transverse momentum distr~utions which in terms of exchange models suggest an equal amount of pion and baryon exchange. For both reactions, we f'md an approximate factorization in the projectile frame between the transverse momentum of nf and the longitudinal momenta of the dissociation products. Exchange models predict this result equally in both reactions, but it appears much more clearly in the A reaction. Finally, we relate the observed longitudinal and transverse momentum distr~utions in a way suggested by an isotropic decay model.
1. Introduction Diffractive production of hadrons [1 ] is by now a well established phenomenon up to ISR energies with a cross section of about a tenth of the total cross section. The most detailed analyses of exclusive diffractive reactions have been carried out * Work supported in part by the U.S. Energy Research and Development Administration under Contract No. AT(I 1-1)-68 for the San Francisco Operations Office. * On leave from Max Planck Institute fur Physik, Munich, Germany. ** Present address: lndiana University, Bloomington, Indiana. * Present address: Iowa State University, Ames, Iowa. ** Present address: University of Pennsylvania, Philadelphia, Pennsylvania. * Present address: CERN, Geneva, Switzerland. ** Present address: DESY, Hamburg, Germany. 405
W. Ochs et al. / Proton diffraction dissociation
406
at lower energies mainly along two lines: (a) the essentially model independent methods of partial-wave analysis, particularly in the resonance region, in order to find the spin-parity states of the dissociated system; (b) comparison of very specific models to the more global features of the data. Most popular are the exchange models of the Drell-Hiida-Deck [2] type, in which the dominance of a single exchange diagram (pion exchange) is supposed. Many modifications of the original ideas have been suggested. One example is the inclusion of nucleon exchange terms [3], and there is now renewed interest in those terms as they may play a role in the explanation of cross-over effects [4]. Other authors [5] have put forward the idea that absorption is important, so that, unlike elastic scattering, diffraction dissociation is peripheral in impact parameter space. This idea leads to some specific predictions on the amplitude structure*, but has not yet been successfully imbedded in a more complete theory to predict the multidimensional distributions of momenta in the final state. In this paper we analyse some global features of the reactions
-*
(1) r.*0r+n)
(2)
at 14 GeV[cin terms of correlations between the Cartesian momentum components of the particles. These reactions have been selected for the present study as there are no resonances in the exotic mr systems and therefore the difficult problem of resonance separation at intermediate energies is avoided. The event sample is then defined simply by a cut in the effective mass of the slow pion and the baryon and in the squared-four-momentum transfer between the beam and the fast pion. More details about the experimental procedure are given in sect. 2. Our choice of variables is explained in sect. 3. The study of momentum correlations is motivated by the observation that exchange models make distinctive predictions on how the transverse momentum of the fast outgoing pion is compensated by the target fragments. In a one pion exchange model for reaction (2) the transverse momentum of the fast pion is compensated by the slow pion, whereas the nucleon has a transverse momentum distribution centered around the origin and the reverse is true for nucleon exchange. These exchange models have the general features of a spectator model and we investigate in sect. 4 how well the data satisfy these features. The importance of transverse momentum correlations as a probe of the underlying spatial structure of the production mechanism has been pointed out some time ago by Blankenbecler and Neff [6]. In their analysis of data, they discussed how one can learn about rescattering effects from two particle correlations. Our analysis on the other hand is more directed to the question of what the underlying exchange mechanisms are. * For example a single zero is expected in the s-channel helicity non-flip amplitudes at - t ~ 0.2 GeV 2. This would show up experimentally at small excitation masses as a single zero of the spherical harmonic moments (Y~) and ( yt> in the s-channel frame.
W. Ochs et aL / Proton diffraction dissociation
407
In sect. 5 we turn to the longitudinal momentum distribution. We study diffraction dissociation in the projectile frame where the initial proton has large momentum. In this frame, within the context of exchange models, the longitudinal momentum distribution of the dissociation fragments is independent of the transverse motion of the scattered projectile and therefore reflects directly the properties of the dissociation vertex. The data of reaction (1) are found to be compatible with this factorization property but there are problems with reaction (2). Though the exchange models considered here are certainly oversimplified, the factorization property is appealing and may be of more general sil;nificance since it is reflected in the data. A presentation of data using our variables is therefore a meaningful alternative to the usual mass and momentum transfer variables where strong correlations exist even in these simplest models. Using spectator model ideas we also try to reconstruct the longitudinal momentum distribution from the transverse momentum distribution. The result is encouraging though not fully satisfactory. In sect. 6 we point out that some of the simple features observed in reactions (1) and (2) are not present in reactions with a strong overlap of the diffractive channel with resonance formation in the mr system. Our conclusions are summarized in sect. 7. 2. The data The data were obtained from a hybrid bubble chamber experiment using the SLAC 40" hydrogen chamber, expanded at up to 12 times/see, and a downstream magnetic spectrometer. The sample of data used in this analysis consists of 6334 lr+p ~ 7r~rr+n events and 4174 7r-p ~ n~-Tr-A++ events. In the restricted kinematic region accepted by the spectrometer, this corresponds to about 22 and 18.5 observed events//~b respectively. The beam momentum for the rr+ exposure was 13.7 GeV/c, and for the rr-, 14.2 GeV/c. The chamber was operated in a triggered mode so that the experiment was only sensitive to events meeting certain kinematic constraints, chosen to select events involving the diffractive dissociation of the proton. Fast forward pions with sufficiently small scattering angle and energy loss could leave the chamber by a thin rear window and enter the magnetic spectrometer. When a scintillator coincidence indicated such an event, magnetostrictive spark chambers were fired and the scattered particle's coordinates in the spectrometer were read into an on-line computer. There, the track was reconstructed and the mass M* of the recoiling system X + in rr±p -~ Ir~X + was calculated. If the event corresponded to an inelastic scatter with M* < 3.8 GeV, the bubble chamber lights were fired. The momentum and direction of the scattered pion were measured with a precision of + 55 MeV/c and + 0.60 mrad, respectively. Further details of the experimental apparatus can be found in ref. [7].
408
w. Ochs et al. / Proton diffraction dissociation
The measurements of the bubble chamber pictures were performed on the SLAC spiral reader, the LBL FSD, and hand measuring machines at both labs. The processing was done with versions of TVGP and SQUAW appropriately modified to include the measurements on the fast pion provided by the spectrometer. The resolutions on the mass-squared of the missing neutral in the one-constraint reactions rr±p ~ rr~pTr0 and 7r~p ~ rr~nTr+ are 0.02 GeV 2 and 0.11 GeV 2, respectively (where the measurements of the n~ track are a hybrid of the bubble chamber and spark chamber measurements). The data have been normalized to elastic scatters observed in untriggered data samples, using published [8] elastic cross sections. The estimated systematic uncertainty in the 7r+n cross sections is +9% and that for n - A ++ is 11%. These estimates include the systematic uncertainty of the elastic cross section in ref. [8]. The errors shown in the figures do not include these systematic uncertainties. For our subsequent study only events out of a restricted kinematic region have been included. This was necessary in order to take care of the finite acceptance of the spectrometer and to minimize the contribution from events in which 7rf instead of n s in reactions (1) and (2) belongs to the diffractive system. All experimental data presented here are subject to the cuts 0.1 ~
(3a)
reaction (1) M*(An) < 3.5 GeV,
(3b)
reaction (2) M*(nlr) < 3.0 GeV,
(3c)
where q is the transverse momentum of the fast pion with respect to the beam. The A events are defined by the mass cut A: 1.12 <~M(prr+) < 1.36 GeV.
(3d)
With this mass cut, the various (Trn) mass distributions show little or no enhancement in the p0 and f0 mass regions. The detection efficiency of the apparatus is a function of M* and q only. In the region defined by the above cuts, 97% of the events were assigned weights less than 4.0. The average weight for events in the n+n sample is 1.70 and that for the 7r-A +÷ is 1.96. The results do not depend significantly on the cut in M*, as the cross sections are reduced by only 7.3% and 5.6%, respectively, if we lower the upper bounds (3b, c) by 500 MeV.
3. Choice of variables Let PB, Pf, P~r, Pn be the momentum vectors of the particles (beam, fast pion, slow pion, neutron) for reaction (2) in the lab frame. We define a Cartesian coordinate system by the unit vectors:
I¢. Ochs et al. / Proton diffraction dissociation
409
z =pB/IPBI, ,v = (PB X p f ) / I P B × p f l ,
(4)
x=y×z, i.e. the beam along z is deflected in the x-direction a n d y is perpendicular to the scattering plane. As independent variables we choose first the three transverse momenta Pfx - q, Pnx and P~ty (and we have Pny = - P ~ y and q = - P ~ x - Pnx)" As the fourth independent variable we take a longitudinal momentum component in the frame where the beam pion is at rest (projectile frame) and the target proton has momentum P = PB " mp/mn "~ 94 GeV/c. The longitudinal momentum fraction x a = Paz/P carried by particle a in that frame is related to the lab variables at high energies (beam velocity ~B ~ 1) by Xa ~ (E a - Paz)/mp.
(5)
For the longitudinal momentum qz of t/'f in that frame we calculate for our low-mass excitation processes
qz "~ q2/2mn + O(1 GeV2/p) .
(6)
The relative longitudinal momentum transfer between the pions, fix = qz/P, becomes negligible * for increasing P and we obtain x,, + x n ,~ 1 .
(7)
We then have essentially only one independent longitudinal momentum and also the kinematic limits 0 < x a < 1 are independent of the particle mass. This contrasts to the situation in the lab frame where the kinematic limits of longitudinal momenta are given by a parabola with the vertex position depending quadratically on the particle mass [9], Infinite momentum frame variables like x a in (5) are used to describe diffraction dissociation in models which treat the nucleon as a composite system [10] and we will see in sect. 5 that the dynamics of exchange models is especially simple in that frame. The conventional choice of variables for a three-body reaction is to take a subenergy of two particles, a momentum transfer t and two internal "decay" angles 0, ¢. Such a choice would be particularly useful if a simple symmetry axis for the differential cross section were present ("helicity conservation") but this is not the case experimentally for any obvious choice of a two-particle subsystem or reference frame **. In the reaction pp + p(nlr +) an asymmetry in the azimuthal distributions in the nn + rest frame has been reported [12], which suggested strong correlations of momenta within the production plane. It is our aim to investigate such correlations in more detail using our variables. * For our data 6x < 0.02. ** For a review see ref. [ 11 ].
410
W. Ochs et al. / Proton diffraction dissociation
4. Study of transverse momentum correlations 4.1. Predictions from simple exchange models We first discuss what kind of correlations are to be expected in simple exchange models. Using a Monte-Carlo technique, we have calculated transverse momentum distributions for the following typical invariant amplitudes describing elementary exchange processes as shown by the diagrams in fig. 1 or combinations thereof, Ta[ -~
s ~r~r on~r eAt tl - ' m 2lr
(8a)
1121 s Tb
~- t l _ m 2 O m r e A t + O l r N t 2 _ r n
r c
z-
~
%
~
+%N
- r n ff2
(Sb)
p2
----
-rn2
+
-
5/*2
,
(8c)
rn
with o~,r/o~rN = 0.6, A = 4 GeV -2, B = 5.5 GeV - 2 and with the momentum transfers t t and subenergies s i as in fig. 1 * We obtain the following results for the transverse momentum distributions of the neutron and the pion for different intervals of the projectile transverse momentum q (fig. 2): model (8aL fig. la, the transverse m o m e n t u m of the fast particle is mainly compensated, on the average, by one particle (the slow pion) whereas the distribution of the other one is centered around the origin, i.e. (Plrx) ~- - q ,
(pnx) ~ 0 .
(9)
This property of model (8a) can be derived analytically in the high-energy limit (see appendix). A corresponding result is obtained for the nucleon exchange diagram of fig. 1b. In model (8b), fig. la, b, where two exchange diagrams are added, the transverse momentum q is compensated by both the neutron and the pion in comparable amounts. Model (8c), figs. la, lb, lc, is similar to model (Sa) at small q due to a partial cancellation of the two nucleon exchange terms. This cancellation is complete atq =0. We also found that the distribution of the m o m e n t u m pny perpendicular to the production plane is independent of q. This is also easily derived analytically (see appendix) for model (8a). Due to these features of transverse momentum correlations we may consider an exchange model like (8a) as a relative realization of a spectator model in which the * These formulae apply for spinless particles; the inclusion of spin is rather complicated in the case of the nucleon Born terms {3] but our results do not depend on them as we have checked for (8a).
W. Ochs et al. I Proton diffraction dissociation
411
t
~-----~
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7r - - - - - - - r - - 7r \
p
n
p
~m
~,f
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"n"
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(b)
(c)
Fig. 1. Born term models for diffraction dissociation.
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- -0.4
0
Fig. 2. Distribution o f the transverse m o m e n t u m components Px in the production plane o f the ~ts and n in ~tp -* nf(nns) at 14 GeV/c, calculated for different intervals o f the transverse momentum q o f Ttf, to demonstrate transverse m o m e n t u m compensation in the production plane in simple Born term models ( S a ) - ( 8 c ) . (a) Single exchange o f fig. la. (b) Superposition o f fig. la and lb. (c) Superposition o f figs. la, l b and lc.
fast particle scatters o f f o n e " c o n s t i t u e n t " , whereas the o t h e r " c o n s t i t u e n t " acts as a s p e c t a t o r and has a transverse m o m e n t u m distribution p e a k e d at the origin.
4.2. Experimental results for momentum correlations in the production plane We first consider the transverse m o m e n t u m distributions o f the s l o w p i o n and
412
W. Ochs et al. /Proton diffraction dissociation 1
1
|
I
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I
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~ - p ~ ~f(~+÷~--) Ot 14 GeV/c
IO0
:t v
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I
1
-I.0
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
Px(GeV/c) Fig. 3. Distributions of the transverse momentum component Px in the production plane for the dissociation fragments ~r- and A++. Data are integrated over the longitudinal variables. A selection M(ZX++~r- ) < 3.5 GeV/c and 0.1 • q • 0.7 GeV/c is applied to all data. The a ++ is defined by 1.12 < M(nn+) • 1.36 GeV.
the baryon after integration over the longitudinal motion. In figs. 3 and 4 we show these distributions for all data of our reactions and in figs. 5 and 6 for some intervals in q. We find the rather surprising regularity that the distributions for the pion and the baryon always have the same shape and in fact are almost identical for q < 0.5 GeVlc. It appears, then, that the slow particles share the transverse momentum transferred by the pomeron in equal parts. This FEnding certainly eliminates models with a single exchange dominating in any region of q, such as models (8a), (8c) above. A model with two exchange diagrams gives better agreement with the data, but the high degree of equality between the contributions from the two terms as shown by figs. 3 - 6 , is very puzzling. This characteristic would only come out of exchange models by a rather artificial interplay of vertex functions or couplings of two amplitudes which are not a pr/or/related in any way. Furthermore, there seems to be no obvious simple explanation of these results in terms of N * resonance production. Whereas there is no doubt about the existence of certain resonances such as the N * (1688) in diffractive channels, the population of spin and helieity states over the full kinematic region is not sufficiently well known that the correlations shown here can be easily explained. Next, we observe a broadening of the distribution in figs. 5 and 6 as q is increased. This may be explained by representing the Px distribution for either particle as a
W. Ochs et el. / Proton diffraction dissociation I
"n'+p~
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413 1
1
"rr~"(n~ + } a t 14 GeV/c
tOO
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::L b~rO
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-0.6
1
-0.4
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-0.2
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0
0.2
0.4
0.6
Px (GeV/c) Fig. 4. Px distribution as in fig. 3 but for the dissociation fragments n and ~r+ with a mass cut M(nn+) < 3 GeV.
superposition of two distributions, one centered at Px = 0, the other one at Px = -q" Furthermore let us assume that these two components are represented by the momentum distribution g(py) of one particle in the y-direction. Then, for fixed q, the Px tlistribution is given by
f(P~rx, q) = f ( P A x' q) = [(g(P~rx) + g(q -- P~rX))h(q) "
0o)
The transverse momentum is compensated on the average by either one o f the two particles. Such a spectator picture can be considered as an abstraction of the exchange models in fig. 1 and eq. (8)~ The curves in fig. 5 are calculated with the experimentally observed distributions g(py) and h(q) of fig. 7, by evaluating eq. (10) in the respective q intervals. For this calculation g ( p y ) has been normalized to unit area. The same procedure has been applied to reaction (2) using the distributions in fig. 8. Though the curves appear to be a little bit broader, the agreement with the data is quite reasonable for q < 0.5 GoV]c. If this picture were correct we would expect a separation of the two peaks at large q, but the data do not show a clear tendency.
4. 3. Distribution perpendicular to the production plane The distribution o f p y in figs. 7 and 8 has been fitted by
W. Ochs et aL / P r o t o n diffraction dissociation
414
I
•
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I
"n'-
~ ' ~ - - - ' r r ( ( ~ ° " n "') ot 14GeV/c
x /3""
I I
100 /
}
_
0"5 < q < 0'7GeV/c I
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~" 1"
i
-
.
.L
i ~
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t t
r
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,
N:,
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[!t I* -08
-0.6
-0.4
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0
0.2
0.4
Px (GeV/c) Fig. 5. Px distribution as in fig. 3 but for intervals in the transverse momentum q of the fast pion. The data suggest that the proton fragments compensate the transverse momentum q in equal parts. Accordingly the curves represent a superposition (eq. (10)) of distributions da[dpy (t'~. 7) centered at Px = 0 and Px = - q , respectively.
do _ C e x p ( _ A x / p 2 + m2)
%
(11)
with the parameters of table 1. The fits are quite reasonable but at py ~ 0.6 GeV/c the data indicate a small break in slope which is not accounted for by (11). As another test of the spectator picture we check whether the motion in the y direction is independent of that in the x direction. In figs. 9 and 10 we show the distribution of py for different cuts in q and P~x. Superimposed on the data points
I¢. Ochs et aL / Proton diffraction dissociation !
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i I I
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~'p--,,-~(r.'rr')ot14GeV/c
• n
I I
I00
o.5
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Ii
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, r1,,
v
b~.
/(
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I00
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o, o o2.v,c , ul ' iX
I
-0.8
-0.6
-0.4
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I 1
0.4
p~ (OeV/c) Fig. 6. Px distribution with data as in fig. 4 but for intervals in the transverse momentum q of the fast pion. The curves again represent the superposition given by eq. (10).
is the same curve (11). Clearly, the data are compatible with having the same distribution. This suggests studying the dependence of do[dpy on both x components Pnx and q simultaneously. This is done in fig. 11 for reaction (2) and again the data are compatible with having the same distributions. We note however, that only the independence Ofpy o f q follows directly from a spectator picture. At q = 0 the common distribution H(P=x, q = O, py) should only depend on p2x + p ; and hence the complete factorization of py,
H(P.x' q' Py) ~ g(Py) f(P.x, q) ,
(12)
as suggested from fig. 11, would imply a Gaussian distribution ofpn x andpy. This
W. Ochs et al. /Proton diffraction dissociation
416
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I¢. Ochs et aL / Proton diffraction dissociation Table 1 Parameters for the fits of
417
do/dpy =C exp(-A #p~ + m2)
Reaction
A(GeV -1)
m(GeV')
x2/d.f.
(1) (2)
7.64 e 0.05 8.89 ~ 0.03
0.18 ± 0.03 0.36 t 0.02
19/15 7/15
in turn is only compatible with the data in figs. 7 and 8 for py <~ 0.4 GeV/c. Therefore the full factorization (12) can be valid only approximately and must be restricted to this py range. It has recently been found [13] that the dispersion Ofpy is independent Ofpx in inclusive pp interactions in a different kinematic regime and therefore eq. (12) may apply to more general situations than diffraction dissociation. The dispersion Ofpy does however depend on the longitudinal variables and, whereas in our data the dispersion varies between 0.1 and 0.3 GeV/c, it varies between 0.2 and 1.0 GeV/c in ref. [13]. i
4Jl
i
/i i
i
,'11
i
i
~'P'-~"tri(,A,++rr-) at 14GeV/c -1.2 < P~r< -0.4GeV/c
-0.2 < p~< 0.0 GeV/c
# ~oo
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~k ,0 ',
,7 ;
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o
lO0
IO o.P
0
0.2
0
0.2
0
0.2
0.4
py (O eV/c)
Fig. 9. Distribution of py (perpendicular to the production plane) for selections of the transverse momenta P~rx and q in the production plane. The data are reasonably represented by the same curve as in fig. 7, appropriately normalized, suggesting the independence of the motion perpendicular to and in the production plane.
418
W. Ochs et aL /Proton diffraction dissociation t
I /'11 1 '~1 rr+P ---~" ",'T~'(n "rr'') o! 14 GeV/c
I
/Jl
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I
}~ -I.2 < p~ < -0.4GeV/c -0.2 < p ~ < 0.0 GeV/c ~ - 0 . 4 < p~<-O.2GeV/c 0.0< O~< 0.8GeV/c
0.1 < q < 0 2 GeV/c 0.3 <_q < 0.5 GeV/c 0.2 < q < 0.3GeV/c 0.5 < cl < 0.7 GeV/c
,oo
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t,% .t~,
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\
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0.2
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0
02
0.4
Fig. 10. Distribution Ofpy as in fig. 9 but for the neutron reaction.
5. The longitudinal motion
5.1. Analysis of the simple exchange models We describe the longitudinal motion using the x variables defined in sect. 3 in the frame in which the dissociating particle has high energy. In terms of the exchange models used above we then have the following picture of the dissociation process. The fast proton first dissociates virtually into a pion and a nucleon. One of these components scatters off the target pion whereby it changes its transverse momentum but not its longitudinal fraction x (see eq. (7)). For pomeron exchange no additional x dependence is introduced at this secondary vertex and the observed x distribution is characteristic of the dissociation vertex itself. Thus we expect a factorization of the form do -
dxdq 2
F(x) G(q2).
(13)
I¢. Ochs et aL / Proton diffraction dissociation -0.2 < p~"<0.0 GeV/c p~>O GeV/c -0.4 501,..~.,} ~
,
419
-1.2 < p~'<-0.4GeV/c < p~'<- 0.2GeV/c
,
%
0
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0
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O La A .Q L. G')
5~ O (.n A
O (30 a)
0
0.2
0.4 "
0.2
0.4 "
0.2
0.4 "
0.2
0.4
py(GeV/c) +
+
÷
Fig. 11. Distribution ofpy in the reaction ~r p--, n Or n) at 14 GeV/c for simultaneous cuts in the transverse m o m e n t a q and P~rx in the production plane. The data are reasonably represented by the same curve as in fig. 8, appropriately normalized, which demonstrates that there is no correlation between the motion perpendicular to and in the production plane for py < 0.4 GeV/c.
This qualitative argument is supported by an explicit calculation for the exchange models in sect. 4 in the high-energy limit, presented in the appendix. We obtain after integration over Pnx and Pry the result
do dx d F
C4q2) =
2 '
,.I/x,,+ r,,2./x.- m p
(14)
and we have the factorization property (I 3). This distribution peaks at x = m~r/(m ~ + ran) which corresponds to the limit where the two particles are free and have the same velocity. Within our approximations the denominator D in (14) may also be written as
420
14/. Ochs et al. /Protondiffraction dissociation
D - m 2 - m~, with m c being the mass of the dissociated system in the case of collinear scattering. Eq. (14) is obtained for both types of exchange processes (nucleon or pion exchange) as expected from our qualitative argument. The cancellation of the exchange poles in the amplitude for collinear scattering was found a long time ago [14] and the above result may be considered as a generalization for q ~ 0. If both types of exchange are present, (14) is valid to the extent that interference effects are negligible. Usually data are presented in terms of the variables q2 (or equivalently tTr~r) and the mass M* of the dissociated system. Clearly, with (12) a strong correlation between q2 and M* results. If we keep x fixed and increase q2 we also increase, say P~rx and therefore M*, so that events with large q2 are found more often with large M*. This correlation has been discussed widely in the past. A likelihood method correlation analysis [14] found disagreement with the prediction of a single exchange model of the above type. For this and other reasons it has been suggested that absorptive effects may be important. Instead, we find it convenient to explore the correlation between x and q2, since a deviation from the factorization property, (13), is easier to detect experimentally, and thus it is easier to test this general feature of the exchange models. We finally mention that the factorization property (13) can be easily generalized to multiparticle production by do/dx I dx 2 ... dq 2 = F(Xl, x 2 ...) G(q2),
(15)
and also to double dissociation. 5.2. Experimental results for the correlation between x and q2 In fig. 12 we show the experimental xTr distributions for the full q2 range and also for selected q2. For the sake of comparison we normalized the distributions to have unit integral. Fig. 13 on the other hand shows the q2 distribution for different x n cuts. The distributions in both reactions are quite similar and show a little shoulder or break at q2 ,,~ 0.2 - 0.3 GeV 2 as is already known from earlier studies
[111. The factorization property (13) works quite well for the A++ reaction, at least for q < 0.5 GeV/c. The data for the highest q bin indicate a slightly faster decrease with x. On the other hand the neutron reaction data do not satisfy (13) so well and the data with q >0.5 are distinctly different from the others. Two effects could cause a violation of (l 3). First there is the presence of nonasymptotic contributions which certainly affect reaction (1), where A+÷ production by p exchange plays a rble [16]. Secondly, the mass cut applied to the data affects the large q data more than the low q data. Both effects should disappear in an experiment at higher beam energy. 5.3. Comparison o f the x distribution with model calculations We compare the x distributions with the following calculations.
IV. Ochs et al. /Proton diffraction dissociation
"n'÷p"~" 7r~'(lr*n) at 14GeV/c
! rr'p"~" uf-(Z~*÷~"') at 14GeV/c • all dato • • 0.1
~x!!ixili!iGeV/c _ "k \\
1.0 ~ , ~ N
0
421
0.5 X.,~
1.0 0
0.5 XTr
1.0
Fig. 12. Distribution (normalized to unit area) of the longitudinal momentum fraction x~r in the projectile frame for all data (0.1 • q • 0.7 GeV/c) and for q selections as indicated. The full ckcles are always at their correct position in x ~ the others are horizontally displaced ff necessary for clarity. The x - q2 factorization hypothesis is quite well satisfied by the A b u t n o t so well by the n reaction. Curves I and II represent exchange model calculations; curve lII is reconstructed using the experimental p.y distribution. The curves are normalized arbitrarily. Also shown is the x n dependence o f the average transverse m o m e n t a Px o f the pion and baryon.
fa) Simple exchange models. The prediction of the exchange models of subsect. 4.1 is given by eq. (14), (curve I). At large x the predicted cross section is too high, in particular for the A reaction. A reduction is obtained if the cross section d a / d p 2 in eq. (A.7) is integrated only up to p±2 = 0.5 GeV 2 instead of to infinity in order to suppress the slow power fall off of the amplitude (curve II). We conclude that the A and n reaction show a dynamical behavior outside the scope of the above models in which the different particles enter only by their masses. We also note that the difference between the two channels cannot be explained by the different charges of the two particles. This can be seen by comparing the channels p --* A++lr - and p --* A07r+, which have the same x dependence for pomeron exchange Osospin I = 0).
422
W. Ochs et al. / Proton diffraction dissociation v
1
~
1
v
1
v
I
v
v
1.0 ~" "n'°p-O"rrt*('/r*n) at 14GeV/c
I
l
1
I
J
1
l
/I- p ~ rrf" (A"'1,r-) Ot 14 GeV/c • oli dolo •
•
0 . 0 < X ~ <0.1
* 0.1 < XTr <0.2 •
*
•
•
0 . 2 < X~r < 0 . 4 0.4
1
1 1
i.c be4
--Ib
OJ
l
0
1
t
0.I
I
,
0.2
I
,
0.3
1
I
0.4
q2 (GeV/c)
,
0.5 0
I
0.1
1
1
0.2
t
1
0.3
L
l
0.4
L
0.5
q2 (GeV/c)
Fig. 13. Distribution (normalized to unit area) of the transverse momentum q2 for different intervals in the longitudinal momentum fraction xTr The full circles are at their correct positions in xn, the others are horizontally displaced if necessary for clarity. Again x - q2 factorization is quite well satisfied by the A++ reaction, but not so well by the n reaction.
(b) Isotropic decay model If we accept the general picture developed at the beginning of this section, we would consider the x n distribution as characteristic of the n-baryon system in the same way as it is for the py distribution studied in the last section. We therefore may try to reconstruct do/dx from da/dpy. This idea of course only makes sense if the factorization (12) holds. We will assume in addition that the pion and the baryon have an isotropic distribution in their rest system for forward scattering, i.e. do = f ( r ) d3r
for
q = 0,
(16)
where r is their rest frame momentum. Clearly for q :/: 0 no isotropy can be expected theoretically within the framework of exchange models as the propagator terms are important, and the angular distributions are also known to be non-isotropic experimentally. A direct test of (16) is possible by examining data at very small q2 (typically q2 < m 2 ) ; production from nuclear targets could possibly provide such data. With assumption (16) we can calculate
do
h ( y ) ==--- = 2rr dpY
f lyl
r drf(r),
(17)
W. Ochs et al. / Proton diffraction dissociation
423
and reconstruct f ( r ) from do/dpy, f(r) -
1
dh(r)
27rr
dr
(18)
'
which yields for our parametric form (11) f(r) -
AC
e x p ( - A v ~ r2 + m 2 ) .
(19)
2rr4r 2 + m 2 Then we finally calculate the desired x,r distribution using
Elf -/'z
xlr -ETr + E ~ '
ER =X/ma2 + r 2 ,
(20)
from the distribution (16). The result of this calculation is represented by curve III. There is disagreement particularly at x~r "" 0, where the calculation predicts a drop whereas the data are finite. The same feature is obtained for curves I and II since eq. (14) implies do/dx n ~ x n at xTr ~ 0. The origin of this zero can be understood easily (see appendix). Since the square 2 ... x~r 2 and phase space cancels only one of the matrix element in (8a) has a term star power ofx~r, we obtain the above linear rise. If we add to the amplitude (8a) a nonasymptotic Regge term like a or fexchange we obtain a constant contribution at xn = 0. If this interpretation is correct the cross section at xTr = 0 should vanish with increasing momentum P like 1/P. 5.4. Transverse m o m e n t u m correlations
We also show in fig. 12 how the transverse momentum q of the pion is compensated by the dissociation products. We learned in the last section that the pion and the baryon have, on the average, the same amount of transverse momentum. In the high-energy limit of the simple exchange models as defined in the appendix, we would then expect ( p n x ) ". (Pnx) " - ~q independent ofx. There is however a clear x dependence of (Px); if a particle has a small longitudinal momentum fraction x, it carries more transverse momentum, i.e. it interacts more strongly with the projectile pion. The fact that (P~x) goes up to zero with increasing x indicates a failure of the simple exchange models and not only a non-asymptotic effect.
6. Investigation of other reactions We also analysed other reactions such as lr+p ~ ~t~(n-A++) ,
(21a)
W. Ochs et al. / Proton diffraction dissociation
424 I
1
1
I
I
I
I
I
I
I
II
Tr+p -. 7r~(ZX++ 7r-)
I000
strait
QTTx
i I
A ++
s
•
I
o °
I I
,t i !
13) L9
tOO
0
U3 Z > i,i i0
-I.0
-0.8
-0.6
I
-o.4
I
-o.z
I
o
1
o z
0.4
0.6
Px (GeV/c)
Fig. 14. Distr~ution of transverse momentum Px as in fig. 3 but for incident *r+. In this reaction ,rn resonance production is important and the similarity of both distributions found in fig. 3 is not met here.
,,-p -,
p),
11"± p ~ lr; (11"0 p ) ,
(21b) (21 C)
from this experiment and also K+d ~ K+Tr- p p s ,
(21 d.)
at 12 GeV/c [ 17]. The distributions of transverse m o m e n t u m Px in these reactions show the same qualitative feature that both the baryon and the meson compensate the transverse m o m e n t u m of the scattered projectile and the Px distributions are quite different from fig. 2a. However the two distributions are no longer so similar as in figs. 3 - 6 . This is shown in fig. 14 for reaction (21a) as a representative example. The reason for this different behavior is the presence of resonance formation in the meson-meson channel which also contributes to the cross section when our mass cuts are applied. We also studied the transverse momentum distributions for the four-body reaction
Ir-p ~ n~-(,r-7r+p) without applying the A cut. We found that the transverse m o m e n t u m components of all the two-particle combinations like P~r-x + P,r+x have the same distribution.
I¢. Ochs et aL / Proton diffraction dissociation
425
However, as there is a competition between different processes at our beam energy, and the diffractive channel is not well separated, we do not go into details here but rather suggest looking for similar regularities at higher beam energies, so that a meaningful interpretation in terms of pomeron exchange is possible. 7. Conclusions
We have analyzed two reactions in the kinematic regime of diffraction dissociation in terms of the Cartesian momentum components of the particle. This study may be considered as complementary to a spin-parity analysis, which places more emphasis on the phenomenon of resonance production. Whereas a spin-parity analysis is especially powerful when it is restricted to certain regions of phase space, our analysis is better suited for finding global features of the considered reactions. The data integrated over the longitudinal motion suggest a spectator picture of the scattering process, as it might be abstracted from the simple exchange processes, in which either one of the two slow particles, but both with the same frequency, takes over the transverse momentum of the scattered projectile. Superimposed on this pattern is an independent relative motion between the two slow particles which is characteristic of the rrN or nA system and is best observed in the direction perpendicular to the production plane. The motion perpendicular to the production plane is found to be independent of the motion in the plane. The degree of similarity (for q < 0.5 GeV/c) between the transverse momentum distributions of the pion and the baryon is however not yet naturally explained by models in which different exchange mechanisms are added, but the observed regularity suggest that such contributions are related in a def'mite way. As expected from exchange models, the longitudinal momentum fractions x of the proton fragments are not correlated with the transverse momentum transferred to the pion target (in the projectile frame) in reaction (1). However some correlation is present in reaction (2). We also tried to extend the above spectator picture to the longitudinal motion and to reconstruct the x~r distribution from that observed for py. This works quite well for large xn. We argued that the disagreement at small x n could be due to nonpomeron exchange and that we expect a dip in do/dxn at x,r "~ 0 to occur at higher energies. Comparing the nucleon and A reaction we observe almost equivalent features in the transverse momentum correlations. On the other hand, the x~r distributfons are quite different and this difference is not reproduced by the exchange models. The same holds for the correlation of ~Px ) with x=. A comparison of the x distributions in various reactions could be most suited for revealing the dynamical characteristics of the various final states after diffraction dissociation. Our analysis is limited in an essential way by the relatively low beam energy, where non-asymptotic effects are still present. A study of momentum correlations
426
I¢. Ochs et aL / Proton diffraction dissociation
in experiments at Fermilab or ISR energies, as they are under way at present, can better determine whether the effects we observe are really characteristic of diffraction dissociation. In particular, it would be interesting to explore, whether or not (and if so, in which kinematic regime) our factorization properties hold at high energies. We are very grateful for enlightening discussions with G.C. Fox. We also wish to thank the SLAC bubble chamber crew, under R. Watt, and the Experimental Facilities Division, in particular L. Keller and J. Murray, for their close cooperation. In addition we thank the Data Analysis groups at our three institutions.
Appendix Here we calculate the distribution of our variables for the amplitude in (8a) (spinless particles), snn F 1/2(q 2)
(A.1)
r=g t I -m 2
In the high-energy limit we assume xP>> 1 GeV and obtain in the projectile frame sitit = 2 x i t P ( E f - qz) '
(A.2)
with the longitudinal momentum qz and energy Ef of the nf being =~ q2 qz -- 2---m '
~2aU2 Ef = (m2n + q2 + '¢z) •
(A.3)
it
Here we neglect terms of order 1 GeV2/p. Furthermore we obtain with p 2 = P2nx + P2ny' t 1 - m 2 =xit(A -P2ni 1 XitXn] '
(A.4)
where A =m 2 p
m2 it
2 mn
Xit
Xrl
The cross section is written as 3 d3pi do : ](2~') 4 ~mITI2 1-[ (2n)32E i i t t ~4(Pa +Pb - P1 - P2 - P3) i---1
(A.5)
W. Ochs et al. / Proton diffraction dissociation
427
To within our approximation, the longitudinal part of the phase space 3
q~L = 5 ( ~ E i - ~/s) 6(~,pz i - P) .i~_l(dPzi/Ei)
(A.6a)
can be evaluated as dx I t
(A.6b)
q~L ~--XnX n E f P " Finally, then
do =
1 32(2rr)4
g2F(q2 ) ( E l - qz )2 m r¢
Ef
dxndq2 dp2n±d~b
(A.7)
2 (,4 - pm/XnX n) 2 x x n '
2 ~ . where we write d2pn.t = l~ dPnld As Pnx enters the formula only quadratically we have
(Pnx) = O,
(p~rx) = - q .
(A.8)
The q dependence can be factored out (note that Ef and qz depend only on q). As a consequence the distribution in the y direction is independent of q. If we integrate over ¢) and p2ni (from 0 to oo) we obtain the factorizing form given in eq. (14). If we perform the same calculation for the nucleon exchange diagram we obtain a formula like (A.7) but with Pnz replaced by p,± and therefore, after integration, we again obtain eq. (14).
References [1] M.L. Good and W.D. Walker, Phys. Rev. 120 (1960) 1857. [2] S.D. Drell and K. Hiida, Phys. Rev. 7 (1961) 199; R.T. Deck, Phys. Rev. Letters 13 (1964) 169. [3] M. Ross and Y.Y. Yam, Phys. Rev. Letters 19 (1967) 546; B.Y. Oh et al., Phys. Rev. D1 (1970) 2494; A. BiaTas,W. Czyz and A. Kotanski, Nucl. Phys. B46 (1972) 109. [4] E.L. Berger, A critique of the reggeized Deck model, ANL-HEP-PR-06 (1975). [5] G. Kane, Acta Phys. Pol. B3 (1972) 845; N. Sakai and J.N.J. White, Nuel. Phys. B59 (1973) 511; S. Humble, Nucl. Phys. B76 (1974) 137; H.I. Miettinen, preprint CERN-TH 1864 (1974). [6] R. Blankenbecler and T.L. Neff, Phys. Rev. D5 (1972) 128. [7] A.R. Dzierba et al., Phys. Rev. D7 (1973) 725; A. Firestone et al., A high statistics triggered bubble chamber experiment to study the low-mass nucleon-pion enhancement formed in pion-nucleon interactions at 14 GeV/c, submitted to 17th Conf. on high-energy physics, London, 1974, and SLAC-PUB in preparation.
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W. Ochs et al. / Proton diffraction dissociation
[8] K.H. Foley et al., Phys. Rev. 181 (1969) 1775. [9] J. Benecke et al., Phys. Rev. 188 (1969) 2159. [10] R. Blankenbecler and S.J. Brodsky, Phys. Rev. D10 (1974) 2973, sect. IIC; L. Resnick, Can. J. Phys. 52 (1974) 2479. [ 11 ] D.W.G.S. Leith, Diffractive processes, Proc. Summer Inst. on particle physics; SLAC report no. 179, vol. 1 (1974). [12] E. Nagy et al., Experimental results on inelastic diffraction scattering in pp collisions at the ISR, submitted to 17th Int. Conf. on high-energy physics, London, 1974. [13] T.S. Clifford et al., Phys. Rev. Letters 34 (1975) 978. [141 L. Stodolsky, Phys. Rev. Letters 18 (1967) 973. [15] H.I. Miettinen and P. Pirila, Phys. Letters 40B (1972) 127. [161 W. Ochs et al., Nucl. Phys. B86 (1975) 253. [17] D. Lissauer et al., Phys. Rev. D6 (1972) 1852.
A STUDY OF TRANSVERSE AND LONGITUDINAL MOMENTUM CORRELATIONS IN PROTON DIFFRACTION DISSOCIATION AT 14 GeV/c *
W. OCHS *, V. DAVIDSON, A. DZIERBA **, A. FIRESTONE ¢, W. FORD **, R. GOMEZ, F. NAGY, C. PECK and C. ROSENFELD California Institute of Technology, Pasadena, California, 91125 J. B A L L A M , J. C A R R O L L , G. C H A D W I C K , D. L I N G L I N * a n d K. M O F F E I T * *
Stanford Linear Accelerator Center, Stanford, California, 94305
R. ELY, D. GRETHER and P. ODDONE Lawrence Berkeley Laboratory, Berkeley, California, 94 720 Received 6 October 1975 We present data on momentum correlations both in and out of the production plane for the reactions n+p--* n~(n+n) and n - p ~ n - 0 r - A ++) The dissociation products f show strikingly similar transverse momentum distr~utions which in terms of exchange models suggest an equal amount of pion and baryon exchange. For both reactions, we f'md an approximate factorization in the projectile frame between the transverse momentum of nf and the longitudinal momenta of the dissociation products. Exchange models predict this result equally in both reactions, but it appears much more clearly in the A reaction. Finally, we relate the observed longitudinal and transverse momentum distr~utions in a way suggested by an isotropic decay model.
1. Introduction Diffractive production of hadrons [1 ] is by now a well established phenomenon up to ISR energies with a cross section of about a tenth of the total cross section. The most detailed analyses of exclusive diffractive reactions have been carried out * Work supported in part by the U.S. Energy Research and Development Administration under Contract No. AT(I 1-1)-68 for the San Francisco Operations Office. * On leave from Max Planck Institute fur Physik, Munich, Germany. ** Present address: lndiana University, Bloomington, Indiana. * Present address: Iowa State University, Ames, Iowa. ** Present address: University of Pennsylvania, Philadelphia, Pennsylvania. * Present address: CERN, Geneva, Switzerland. ** Present address: DESY, Hamburg, Germany. 405
W. Ochs et al. / Proton diffraction dissociation
406
at lower energies mainly along two lines: (a) the essentially model independent methods of partial-wave analysis, particularly in the resonance region, in order to find the spin-parity states of the dissociated system; (b) comparison of very specific models to the more global features of the data. Most popular are the exchange models of the Drell-Hiida-Deck [2] type, in which the dominance of a single exchange diagram (pion exchange) is supposed. Many modifications of the original ideas have been suggested. One example is the inclusion of nucleon exchange terms [3], and there is now renewed interest in those terms as they may play a role in the explanation of cross-over effects [4]. Other authors [5] have put forward the idea that absorption is important, so that, unlike elastic scattering, diffraction dissociation is peripheral in impact parameter space. This idea leads to some specific predictions on the amplitude structure*, but has not yet been successfully imbedded in a more complete theory to predict the multidimensional distributions of momenta in the final state. In this paper we analyse some global features of the reactions
-*
(1) r.*0r+n)
(2)
at 14 GeV[cin terms of correlations between the Cartesian momentum components of the particles. These reactions have been selected for the present study as there are no resonances in the exotic mr systems and therefore the difficult problem of resonance separation at intermediate energies is avoided. The event sample is then defined simply by a cut in the effective mass of the slow pion and the baryon and in the squared-four-momentum transfer between the beam and the fast pion. More details about the experimental procedure are given in sect. 2. Our choice of variables is explained in sect. 3. The study of momentum correlations is motivated by the observation that exchange models make distinctive predictions on how the transverse momentum of the fast outgoing pion is compensated by the target fragments. In a one pion exchange model for reaction (2) the transverse momentum of the fast pion is compensated by the slow pion, whereas the nucleon has a transverse momentum distribution centered around the origin and the reverse is true for nucleon exchange. These exchange models have the general features of a spectator model and we investigate in sect. 4 how well the data satisfy these features. The importance of transverse momentum correlations as a probe of the underlying spatial structure of the production mechanism has been pointed out some time ago by Blankenbecler and Neff [6]. In their analysis of data, they discussed how one can learn about rescattering effects from two particle correlations. Our analysis on the other hand is more directed to the question of what the underlying exchange mechanisms are. * For example a single zero is expected in the s-channel helicity non-flip amplitudes at - t ~ 0.2 GeV 2. This would show up experimentally at small excitation masses as a single zero of the spherical harmonic moments (Y~) and ( yt> in the s-channel frame.
W. Ochs et aL / Proton diffraction dissociation
407
In sect. 5 we turn to the longitudinal momentum distribution. We study diffraction dissociation in the projectile frame where the initial proton has large momentum. In this frame, within the context of exchange models, the longitudinal momentum distribution of the dissociation fragments is independent of the transverse motion of the scattered projectile and therefore reflects directly the properties of the dissociation vertex. The data of reaction (1) are found to be compatible with this factorization property but there are problems with reaction (2). Though the exchange models considered here are certainly oversimplified, the factorization property is appealing and may be of more general sil;nificance since it is reflected in the data. A presentation of data using our variables is therefore a meaningful alternative to the usual mass and momentum transfer variables where strong correlations exist even in these simplest models. Using spectator model ideas we also try to reconstruct the longitudinal momentum distribution from the transverse momentum distribution. The result is encouraging though not fully satisfactory. In sect. 6 we point out that some of the simple features observed in reactions (1) and (2) are not present in reactions with a strong overlap of the diffractive channel with resonance formation in the mr system. Our conclusions are summarized in sect. 7. 2. The data The data were obtained from a hybrid bubble chamber experiment using the SLAC 40" hydrogen chamber, expanded at up to 12 times/see, and a downstream magnetic spectrometer. The sample of data used in this analysis consists of 6334 lr+p ~ 7r~rr+n events and 4174 7r-p ~ n~-Tr-A++ events. In the restricted kinematic region accepted by the spectrometer, this corresponds to about 22 and 18.5 observed events//~b respectively. The beam momentum for the rr+ exposure was 13.7 GeV/c, and for the rr-, 14.2 GeV/c. The chamber was operated in a triggered mode so that the experiment was only sensitive to events meeting certain kinematic constraints, chosen to select events involving the diffractive dissociation of the proton. Fast forward pions with sufficiently small scattering angle and energy loss could leave the chamber by a thin rear window and enter the magnetic spectrometer. When a scintillator coincidence indicated such an event, magnetostrictive spark chambers were fired and the scattered particle's coordinates in the spectrometer were read into an on-line computer. There, the track was reconstructed and the mass M* of the recoiling system X + in rr±p -~ Ir~X + was calculated. If the event corresponded to an inelastic scatter with M* < 3.8 GeV, the bubble chamber lights were fired. The momentum and direction of the scattered pion were measured with a precision of + 55 MeV/c and + 0.60 mrad, respectively. Further details of the experimental apparatus can be found in ref. [7].
408
w. Ochs et al. / Proton diffraction dissociation
The measurements of the bubble chamber pictures were performed on the SLAC spiral reader, the LBL FSD, and hand measuring machines at both labs. The processing was done with versions of TVGP and SQUAW appropriately modified to include the measurements on the fast pion provided by the spectrometer. The resolutions on the mass-squared of the missing neutral in the one-constraint reactions rr±p ~ rr~pTr0 and 7r~p ~ rr~nTr+ are 0.02 GeV 2 and 0.11 GeV 2, respectively (where the measurements of the n~ track are a hybrid of the bubble chamber and spark chamber measurements). The data have been normalized to elastic scatters observed in untriggered data samples, using published [8] elastic cross sections. The estimated systematic uncertainty in the 7r+n cross sections is +9% and that for n - A ++ is 11%. These estimates include the systematic uncertainty of the elastic cross section in ref. [8]. The errors shown in the figures do not include these systematic uncertainties. For our subsequent study only events out of a restricted kinematic region have been included. This was necessary in order to take care of the finite acceptance of the spectrometer and to minimize the contribution from events in which 7rf instead of n s in reactions (1) and (2) belongs to the diffractive system. All experimental data presented here are subject to the cuts 0.1 ~
(3a)
reaction (1) M*(An) < 3.5 GeV,
(3b)
reaction (2) M*(nlr) < 3.0 GeV,
(3c)
where q is the transverse momentum of the fast pion with respect to the beam. The A events are defined by the mass cut A: 1.12 <~M(prr+) < 1.36 GeV.
(3d)
With this mass cut, the various (Trn) mass distributions show little or no enhancement in the p0 and f0 mass regions. The detection efficiency of the apparatus is a function of M* and q only. In the region defined by the above cuts, 97% of the events were assigned weights less than 4.0. The average weight for events in the n+n sample is 1.70 and that for the 7r-A +÷ is 1.96. The results do not depend significantly on the cut in M*, as the cross sections are reduced by only 7.3% and 5.6%, respectively, if we lower the upper bounds (3b, c) by 500 MeV.
3. Choice of variables Let PB, Pf, P~r, Pn be the momentum vectors of the particles (beam, fast pion, slow pion, neutron) for reaction (2) in the lab frame. We define a Cartesian coordinate system by the unit vectors:
I¢. Ochs et al. / Proton diffraction dissociation
409
z =pB/IPBI, ,v = (PB X p f ) / I P B × p f l ,
(4)
x=y×z, i.e. the beam along z is deflected in the x-direction a n d y is perpendicular to the scattering plane. As independent variables we choose first the three transverse momenta Pfx - q, Pnx and P~ty (and we have Pny = - P ~ y and q = - P ~ x - Pnx)" As the fourth independent variable we take a longitudinal momentum component in the frame where the beam pion is at rest (projectile frame) and the target proton has momentum P = PB " mp/mn "~ 94 GeV/c. The longitudinal momentum fraction x a = Paz/P carried by particle a in that frame is related to the lab variables at high energies (beam velocity ~B ~ 1) by Xa ~ (E a - Paz)/mp.
(5)
For the longitudinal momentum qz of t/'f in that frame we calculate for our low-mass excitation processes
qz "~ q2/2mn + O(1 GeV2/p) .
(6)
The relative longitudinal momentum transfer between the pions, fix = qz/P, becomes negligible * for increasing P and we obtain x,, + x n ,~ 1 .
(7)
We then have essentially only one independent longitudinal momentum and also the kinematic limits 0 < x a < 1 are independent of the particle mass. This contrasts to the situation in the lab frame where the kinematic limits of longitudinal momenta are given by a parabola with the vertex position depending quadratically on the particle mass [9], Infinite momentum frame variables like x a in (5) are used to describe diffraction dissociation in models which treat the nucleon as a composite system [10] and we will see in sect. 5 that the dynamics of exchange models is especially simple in that frame. The conventional choice of variables for a three-body reaction is to take a subenergy of two particles, a momentum transfer t and two internal "decay" angles 0, ¢. Such a choice would be particularly useful if a simple symmetry axis for the differential cross section were present ("helicity conservation") but this is not the case experimentally for any obvious choice of a two-particle subsystem or reference frame **. In the reaction pp + p(nlr +) an asymmetry in the azimuthal distributions in the nn + rest frame has been reported [12], which suggested strong correlations of momenta within the production plane. It is our aim to investigate such correlations in more detail using our variables. * For our data 6x < 0.02. ** For a review see ref. [ 11 ].
410
W. Ochs et al. / Proton diffraction dissociation
4. Study of transverse momentum correlations 4.1. Predictions from simple exchange models We first discuss what kind of correlations are to be expected in simple exchange models. Using a Monte-Carlo technique, we have calculated transverse momentum distributions for the following typical invariant amplitudes describing elementary exchange processes as shown by the diagrams in fig. 1 or combinations thereof, Ta[ -~
s ~r~r on~r eAt tl - ' m 2lr
(8a)
1121 s Tb
~- t l _ m 2 O m r e A t + O l r N t 2 _ r n
r c
z-
~
%
~
+%N
- r n ff2
(Sb)
p2
----
-rn2
+
-
5/*2
,
(8c)
rn
with o~,r/o~rN = 0.6, A = 4 GeV -2, B = 5.5 GeV - 2 and with the momentum transfers t t and subenergies s i as in fig. 1 * We obtain the following results for the transverse momentum distributions of the neutron and the pion for different intervals of the projectile transverse momentum q (fig. 2): model (8aL fig. la, the transverse m o m e n t u m of the fast particle is mainly compensated, on the average, by one particle (the slow pion) whereas the distribution of the other one is centered around the origin, i.e. (Plrx) ~- - q ,
(pnx) ~ 0 .
(9)
This property of model (8a) can be derived analytically in the high-energy limit (see appendix). A corresponding result is obtained for the nucleon exchange diagram of fig. 1b. In model (8b), fig. la, b, where two exchange diagrams are added, the transverse momentum q is compensated by both the neutron and the pion in comparable amounts. Model (8c), figs. la, lb, lc, is similar to model (Sa) at small q due to a partial cancellation of the two nucleon exchange terms. This cancellation is complete atq =0. We also found that the distribution of the m o m e n t u m pny perpendicular to the production plane is independent of q. This is also easily derived analytically (see appendix) for model (8a). Due to these features of transverse momentum correlations we may consider an exchange model like (8a) as a relative realization of a spectator model in which the * These formulae apply for spinless particles; the inclusion of spin is rather complicated in the case of the nucleon Born terms {3] but our results do not depend on them as we have checked for (8a).
W. Ochs et al. I Proton diffraction dissociation
411
t
~-----~
7r ~
7r - - - - - - - r - - 7r \
p
n
p
~m
~,f
tl
p \
"n"
¢
p~n)M
t2
(o)
(b)
(c)
Fig. 1. Born term models for diffraction dissociation.
',
' I"
ot
":"'
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";. . . .
r ' '!
o.z
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/V', /,'k~\
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~. /i\
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(c)
'\\
~.
V
I0
-0.4
0
"-0.4
0 -'0.8 -0.4 0 Px (GeV/c)
- -0.4
0
Fig. 2. Distribution o f the transverse m o m e n t u m components Px in the production plane o f the ~ts and n in ~tp -* nf(nns) at 14 GeV/c, calculated for different intervals o f the transverse momentum q o f Ttf, to demonstrate transverse m o m e n t u m compensation in the production plane in simple Born term models ( S a ) - ( 8 c ) . (a) Single exchange o f fig. la. (b) Superposition o f fig. la and lb. (c) Superposition o f figs. la, l b and lc.
fast particle scatters o f f o n e " c o n s t i t u e n t " , whereas the o t h e r " c o n s t i t u e n t " acts as a s p e c t a t o r and has a transverse m o m e n t u m distribution p e a k e d at the origin.
4.2. Experimental results for momentum correlations in the production plane We first consider the transverse m o m e n t u m distributions o f the s l o w p i o n and
412
W. Ochs et al. /Proton diffraction dissociation 1
1
|
I
I
1
I
1
~ - p ~ ~f(~+÷~--) Ot 14 GeV/c
IO0
:t v
I
I
1
I
I
I
I
1
-I.0
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
Px(GeV/c) Fig. 3. Distributions of the transverse momentum component Px in the production plane for the dissociation fragments ~r- and A++. Data are integrated over the longitudinal variables. A selection M(ZX++~r- ) < 3.5 GeV/c and 0.1 • q • 0.7 GeV/c is applied to all data. The a ++ is defined by 1.12 < M(nn+) • 1.36 GeV.
the baryon after integration over the longitudinal motion. In figs. 3 and 4 we show these distributions for all data of our reactions and in figs. 5 and 6 for some intervals in q. We find the rather surprising regularity that the distributions for the pion and the baryon always have the same shape and in fact are almost identical for q < 0.5 GeVlc. It appears, then, that the slow particles share the transverse momentum transferred by the pomeron in equal parts. This FEnding certainly eliminates models with a single exchange dominating in any region of q, such as models (8a), (8c) above. A model with two exchange diagrams gives better agreement with the data, but the high degree of equality between the contributions from the two terms as shown by figs. 3 - 6 , is very puzzling. This characteristic would only come out of exchange models by a rather artificial interplay of vertex functions or couplings of two amplitudes which are not a pr/or/related in any way. Furthermore, there seems to be no obvious simple explanation of these results in terms of N * resonance production. Whereas there is no doubt about the existence of certain resonances such as the N * (1688) in diffractive channels, the population of spin and helieity states over the full kinematic region is not sufficiently well known that the correlations shown here can be easily explained. Next, we observe a broadening of the distribution in figs. 5 and 6 as q is increased. This may be explained by representing the Px distribution for either particle as a
W. Ochs et el. / Proton diffraction dissociation I
"n'+p~
I
l
I
I
I
I
413 1
1
"rr~"(n~ + } a t 14 GeV/c
tOO
(D
::L b~rO
1
I
-I.0
-0.8
1
-0.6
1
-0.4
I
-0.2
I
I
I
J
0
0.2
0.4
0.6
Px (GeV/c) Fig. 4. Px distribution as in fig. 3 but for the dissociation fragments n and ~r+ with a mass cut M(nn+) < 3 GeV.
superposition of two distributions, one centered at Px = 0, the other one at Px = -q" Furthermore let us assume that these two components are represented by the momentum distribution g(py) of one particle in the y-direction. Then, for fixed q, the Px tlistribution is given by
f(P~rx, q) = f ( P A x' q) = [(g(P~rx) + g(q -- P~rX))h(q) "
0o)
The transverse momentum is compensated on the average by either one o f the two particles. Such a spectator picture can be considered as an abstraction of the exchange models in fig. 1 and eq. (8)~ The curves in fig. 5 are calculated with the experimentally observed distributions g(py) and h(q) of fig. 7, by evaluating eq. (10) in the respective q intervals. For this calculation g ( p y ) has been normalized to unit area. The same procedure has been applied to reaction (2) using the distributions in fig. 8. Though the curves appear to be a little bit broader, the agreement with the data is quite reasonable for q < 0.5 GoV]c. If this picture were correct we would expect a separation of the two peaks at large q, but the data do not show a clear tendency.
4. 3. Distribution perpendicular to the production plane The distribution o f p y in figs. 7 and 8 has been fitted by
W. Ochs et aL / P r o t o n diffraction dissociation
414
I
•
I
I
i
I
I
"n'-
~ ' ~ - - - ' r r ( ( ~ ° " n "') ot 14GeV/c
x /3""
I I
100 /
}
_
0"5 < q < 0'7GeV/c I
Zt
~ 100
~" 1"
i
-
.
.L
i ~
/'
t t
r
i,'} ~ j
,o
,
N:,
,
[!t I* -08
-0.6
-0.4
~t -0.2
0
0.2
0.4
Px (GeV/c) Fig. 5. Px distribution as in fig. 3 but for intervals in the transverse momentum q of the fast pion. The data suggest that the proton fragments compensate the transverse momentum q in equal parts. Accordingly the curves represent a superposition (eq. (10)) of distributions da[dpy (t'~. 7) centered at Px = 0 and Px = - q , respectively.
do _ C e x p ( _ A x / p 2 + m2)
%
(11)
with the parameters of table 1. The fits are quite reasonable but at py ~ 0.6 GeV/c the data indicate a small break in slope which is not accounted for by (11). As another test of the spectator picture we check whether the motion in the y direction is independent of that in the x direction. In figs. 9 and 10 we show the distribution of py for different cuts in q and P~x. Superimposed on the data points
I¢. Ochs et aL / Proton diffraction dissociation !
I
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• ~"
i I I
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415
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~'p--,,-~(r.'rr')ot14GeV/c
• n
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"•100
.~'~ 0.3< q < 0.5 GeV/ci "t:\t. t_
Ii
, ! x~
, r1,,
v
b~.
/(
I
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I00
,
o, o o2.v,c , ul ' iX
I
-0.8
-0.6
-0.4
I
-0.2
1
0
O.2
I 1
0.4
p~ (OeV/c) Fig. 6. Px distribution with data as in fig. 4 but for intervals in the transverse momentum q of the fast pion. The curves again represent the superposition given by eq. (10).
is the same curve (11). Clearly, the data are compatible with having the same distribution. This suggests studying the dependence of do[dpy on both x components Pnx and q simultaneously. This is done in fig. 11 for reaction (2) and again the data are compatible with having the same distributions. We note however, that only the independence Ofpy o f q follows directly from a spectator picture. At q = 0 the common distribution H(P=x, q = O, py) should only depend on p2x + p ; and hence the complete factorization of py,
H(P.x' q' Py) ~ g(Py) f(P.x, q) ,
(12)
as suggested from fig. 11, would imply a Gaussian distribution ofpn x andpy. This
W. Ochs et al. /Proton diffraction dissociation
416
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(~/AaOI/Q~
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,.,
I¢. Ochs et aL / Proton diffraction dissociation Table 1 Parameters for the fits of
417
do/dpy =C exp(-A #p~ + m2)
Reaction
A(GeV -1)
m(GeV')
x2/d.f.
(1) (2)
7.64 e 0.05 8.89 ~ 0.03
0.18 ± 0.03 0.36 t 0.02
19/15 7/15
in turn is only compatible with the data in figs. 7 and 8 for py <~ 0.4 GeV/c. Therefore the full factorization (12) can be valid only approximately and must be restricted to this py range. It has recently been found [13] that the dispersion Ofpy is independent Ofpx in inclusive pp interactions in a different kinematic regime and therefore eq. (12) may apply to more general situations than diffraction dissociation. The dispersion Ofpy does however depend on the longitudinal variables and, whereas in our data the dispersion varies between 0.1 and 0.3 GeV/c, it varies between 0.2 and 1.0 GeV/c in ref. [13]. i
4Jl
i
/i i
i
,'11
i
i
~'P'-~"tri(,A,++rr-) at 14GeV/c -1.2 < P~r< -0.4GeV/c
-0.2 < p~< 0.0 GeV/c
# ~oo
'~
~k ,0 ',
,7 ;
t
I
03 < q < 0.2 GeV/c 0.3 < q < 0.5 GeV/c 0.2
o
lO0
IO o.P
0
0.2
0
0.2
0
0.2
0.4
py (O eV/c)
Fig. 9. Distribution of py (perpendicular to the production plane) for selections of the transverse momenta P~rx and q in the production plane. The data are reasonably represented by the same curve as in fig. 7, appropriately normalized, suggesting the independence of the motion perpendicular to and in the production plane.
418
W. Ochs et aL /Proton diffraction dissociation t
I /'11 1 '~1 rr+P ---~" ",'T~'(n "rr'') o! 14 GeV/c
I
/Jl
[
I
}~ -I.2 < p~ < -0.4GeV/c -0.2 < p ~ < 0.0 GeV/c ~ - 0 . 4 < p~<-O.2GeV/c 0.0< O~< 0.8GeV/c
0.1 < q < 0 2 GeV/c 0.3 <_q < 0.5 GeV/c 0.2 < q < 0.3GeV/c 0.5 < cl < 0.7 GeV/c
,oo
.,.~
t,% .t~,
"~,.,,~
,,, \
\
x
,G,
\-%
k
0.2
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OZ
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0
02
0.4
Fig. 10. Distribution Ofpy as in fig. 9 but for the neutron reaction.
5. The longitudinal motion
5.1. Analysis of the simple exchange models We describe the longitudinal motion using the x variables defined in sect. 3 in the frame in which the dissociating particle has high energy. In terms of the exchange models used above we then have the following picture of the dissociation process. The fast proton first dissociates virtually into a pion and a nucleon. One of these components scatters off the target pion whereby it changes its transverse momentum but not its longitudinal fraction x (see eq. (7)). For pomeron exchange no additional x dependence is introduced at this secondary vertex and the observed x distribution is characteristic of the dissociation vertex itself. Thus we expect a factorization of the form do -
dxdq 2
F(x) G(q2).
(13)
I¢. Ochs et aL / Proton diffraction dissociation -0.2 < p~"<0.0 GeV/c p~>O GeV/c -0.4 501,..~.,} ~
,
419
-1.2 < p~'<-0.4GeV/c < p~'<- 0.2GeV/c
,
%
0
^
0
P t~ A A P ,5")
O La A .Q L. G')
5~ O (.n A
O (30 a)
0
0.2
0.4 "
0.2
0.4 "
0.2
0.4 "
0.2
0.4
py(GeV/c) +
+
÷
Fig. 11. Distribution ofpy in the reaction ~r p--, n Or n) at 14 GeV/c for simultaneous cuts in the transverse m o m e n t a q and P~rx in the production plane. The data are reasonably represented by the same curve as in fig. 8, appropriately normalized, which demonstrates that there is no correlation between the motion perpendicular to and in the production plane for py < 0.4 GeV/c.
This qualitative argument is supported by an explicit calculation for the exchange models in sect. 4 in the high-energy limit, presented in the appendix. We obtain after integration over Pnx and Pry the result
do dx d F
C4q2) =
2 '
,.I/x,,+ r,,2./x.- m p
(14)
and we have the factorization property (I 3). This distribution peaks at x = m~r/(m ~ + ran) which corresponds to the limit where the two particles are free and have the same velocity. Within our approximations the denominator D in (14) may also be written as
420
14/. Ochs et al. /Protondiffraction dissociation
D - m 2 - m~, with m c being the mass of the dissociated system in the case of collinear scattering. Eq. (14) is obtained for both types of exchange processes (nucleon or pion exchange) as expected from our qualitative argument. The cancellation of the exchange poles in the amplitude for collinear scattering was found a long time ago [14] and the above result may be considered as a generalization for q ~ 0. If both types of exchange are present, (14) is valid to the extent that interference effects are negligible. Usually data are presented in terms of the variables q2 (or equivalently tTr~r) and the mass M* of the dissociated system. Clearly, with (12) a strong correlation between q2 and M* results. If we keep x fixed and increase q2 we also increase, say P~rx and therefore M*, so that events with large q2 are found more often with large M*. This correlation has been discussed widely in the past. A likelihood method correlation analysis [14] found disagreement with the prediction of a single exchange model of the above type. For this and other reasons it has been suggested that absorptive effects may be important. Instead, we find it convenient to explore the correlation between x and q2, since a deviation from the factorization property, (13), is easier to detect experimentally, and thus it is easier to test this general feature of the exchange models. We finally mention that the factorization property (13) can be easily generalized to multiparticle production by do/dx I dx 2 ... dq 2 = F(Xl, x 2 ...) G(q2),
(15)
and also to double dissociation. 5.2. Experimental results for the correlation between x and q2 In fig. 12 we show the experimental xTr distributions for the full q2 range and also for selected q2. For the sake of comparison we normalized the distributions to have unit integral. Fig. 13 on the other hand shows the q2 distribution for different x n cuts. The distributions in both reactions are quite similar and show a little shoulder or break at q2 ,,~ 0.2 - 0.3 GeV 2 as is already known from earlier studies
[111. The factorization property (13) works quite well for the A++ reaction, at least for q < 0.5 GeV/c. The data for the highest q bin indicate a slightly faster decrease with x. On the other hand the neutron reaction data do not satisfy (13) so well and the data with q >0.5 are distinctly different from the others. Two effects could cause a violation of (l 3). First there is the presence of nonasymptotic contributions which certainly affect reaction (1), where A+÷ production by p exchange plays a rble [16]. Secondly, the mass cut applied to the data affects the large q data more than the low q data. Both effects should disappear in an experiment at higher beam energy. 5.3. Comparison o f the x distribution with model calculations We compare the x distributions with the following calculations.
IV. Ochs et al. /Proton diffraction dissociation
"n'÷p"~" 7r~'(lr*n) at 14GeV/c
! rr'p"~" uf-(Z~*÷~"') at 14GeV/c • all dato • • 0.1
~x!!ixili!iGeV/c _ "k \\
1.0 ~ , ~ N
0
421
0.5 X.,~
1.0 0
0.5 XTr
1.0
Fig. 12. Distribution (normalized to unit area) of the longitudinal momentum fraction x~r in the projectile frame for all data (0.1 • q • 0.7 GeV/c) and for q selections as indicated. The full ckcles are always at their correct position in x ~ the others are horizontally displaced ff necessary for clarity. The x - q2 factorization hypothesis is quite well satisfied by the A b u t n o t so well by the n reaction. Curves I and II represent exchange model calculations; curve lII is reconstructed using the experimental p.y distribution. The curves are normalized arbitrarily. Also shown is the x n dependence o f the average transverse m o m e n t a Px o f the pion and baryon.
fa) Simple exchange models. The prediction of the exchange models of subsect. 4.1 is given by eq. (14), (curve I). At large x the predicted cross section is too high, in particular for the A reaction. A reduction is obtained if the cross section d a / d p 2 in eq. (A.7) is integrated only up to p±2 = 0.5 GeV 2 instead of to infinity in order to suppress the slow power fall off of the amplitude (curve II). We conclude that the A and n reaction show a dynamical behavior outside the scope of the above models in which the different particles enter only by their masses. We also note that the difference between the two channels cannot be explained by the different charges of the two particles. This can be seen by comparing the channels p --* A++lr - and p --* A07r+, which have the same x dependence for pomeron exchange Osospin I = 0).
422
W. Ochs et al. / Proton diffraction dissociation v
1
~
1
v
1
v
I
v
v
1.0 ~" "n'°p-O"rrt*('/r*n) at 14GeV/c
I
l
1
I
J
1
l
/I- p ~ rrf" (A"'1,r-) Ot 14 GeV/c • oli dolo •
•
0 . 0 < X ~ <0.1
* 0.1 < XTr <0.2 •
*
•
•
0 . 2 < X~r < 0 . 4 0.4
1
1 1
i.c be4
--Ib
OJ
l
0
1
t
0.I
I
,
0.2
I
,
0.3
1
I
0.4
q2 (GeV/c)
,
0.5 0
I
0.1
1
1
0.2
t
1
0.3
L
l
0.4
L
0.5
q2 (GeV/c)
Fig. 13. Distribution (normalized to unit area) of the transverse momentum q2 for different intervals in the longitudinal momentum fraction xTr The full circles are at their correct positions in xn, the others are horizontally displaced if necessary for clarity. Again x - q2 factorization is quite well satisfied by the A++ reaction, but not so well by the n reaction.
(b) Isotropic decay model If we accept the general picture developed at the beginning of this section, we would consider the x n distribution as characteristic of the n-baryon system in the same way as it is for the py distribution studied in the last section. We therefore may try to reconstruct do/dx from da/dpy. This idea of course only makes sense if the factorization (12) holds. We will assume in addition that the pion and the baryon have an isotropic distribution in their rest system for forward scattering, i.e. do = f ( r ) d3r
for
q = 0,
(16)
where r is their rest frame momentum. Clearly for q :/: 0 no isotropy can be expected theoretically within the framework of exchange models as the propagator terms are important, and the angular distributions are also known to be non-isotropic experimentally. A direct test of (16) is possible by examining data at very small q2 (typically q2 < m 2 ) ; production from nuclear targets could possibly provide such data. With assumption (16) we can calculate
do
h ( y ) ==--- = 2rr dpY
f lyl
r drf(r),
(17)
W. Ochs et al. / Proton diffraction dissociation
423
and reconstruct f ( r ) from do/dpy, f(r) -
1
dh(r)
27rr
dr
(18)
'
which yields for our parametric form (11) f(r) -
AC
e x p ( - A v ~ r2 + m 2 ) .
(19)
2rr4r 2 + m 2 Then we finally calculate the desired x,r distribution using
Elf -/'z
xlr -ETr + E ~ '
ER =X/ma2 + r 2 ,
(20)
from the distribution (16). The result of this calculation is represented by curve III. There is disagreement particularly at x~r "" 0, where the calculation predicts a drop whereas the data are finite. The same feature is obtained for curves I and II since eq. (14) implies do/dx n ~ x n at xTr ~ 0. The origin of this zero can be understood easily (see appendix). Since the square 2 ... x~r 2 and phase space cancels only one of the matrix element in (8a) has a term star power ofx~r, we obtain the above linear rise. If we add to the amplitude (8a) a nonasymptotic Regge term like a or fexchange we obtain a constant contribution at xn = 0. If this interpretation is correct the cross section at xTr = 0 should vanish with increasing momentum P like 1/P. 5.4. Transverse m o m e n t u m correlations
We also show in fig. 12 how the transverse momentum q of the pion is compensated by the dissociation products. We learned in the last section that the pion and the baryon have, on the average, the same amount of transverse momentum. In the high-energy limit of the simple exchange models as defined in the appendix, we would then expect ( p n x ) ". (Pnx) " - ~q independent ofx. There is however a clear x dependence of (Px); if a particle has a small longitudinal momentum fraction x, it carries more transverse momentum, i.e. it interacts more strongly with the projectile pion. The fact that (P~x) goes up to zero with increasing x indicates a failure of the simple exchange models and not only a non-asymptotic effect.
6. Investigation of other reactions We also analysed other reactions such as lr+p ~ ~t~(n-A++) ,
(21a)
W. Ochs et al. / Proton diffraction dissociation
424 I
1
1
I
I
I
I
I
I
I
II
Tr+p -. 7r~(ZX++ 7r-)
I000
strait
QTTx
i I
A ++
s
•
I
o °
I I
,t i !
13) L9
tOO
0
U3 Z > i,i i0
-I.0
-0.8
-0.6
I
-o.4
I
-o.z
I
o
1
o z
0.4
0.6
Px (GeV/c)
Fig. 14. Distr~ution of transverse momentum Px as in fig. 3 but for incident *r+. In this reaction ,rn resonance production is important and the similarity of both distributions found in fig. 3 is not met here.
,,-p -,
p),
11"± p ~ lr; (11"0 p ) ,
(21b) (21 C)
from this experiment and also K+d ~ K+Tr- p p s ,
(21 d.)
at 12 GeV/c [ 17]. The distributions of transverse m o m e n t u m Px in these reactions show the same qualitative feature that both the baryon and the meson compensate the transverse m o m e n t u m of the scattered projectile and the Px distributions are quite different from fig. 2a. However the two distributions are no longer so similar as in figs. 3 - 6 . This is shown in fig. 14 for reaction (21a) as a representative example. The reason for this different behavior is the presence of resonance formation in the meson-meson channel which also contributes to the cross section when our mass cuts are applied. We also studied the transverse momentum distributions for the four-body reaction
Ir-p ~ n~-(,r-7r+p) without applying the A cut. We found that the transverse m o m e n t u m components of all the two-particle combinations like P~r-x + P,r+x have the same distribution.
I¢. Ochs et aL / Proton diffraction dissociation
425
However, as there is a competition between different processes at our beam energy, and the diffractive channel is not well separated, we do not go into details here but rather suggest looking for similar regularities at higher beam energies, so that a meaningful interpretation in terms of pomeron exchange is possible. 7. Conclusions
We have analyzed two reactions in the kinematic regime of diffraction dissociation in terms of the Cartesian momentum components of the particle. This study may be considered as complementary to a spin-parity analysis, which places more emphasis on the phenomenon of resonance production. Whereas a spin-parity analysis is especially powerful when it is restricted to certain regions of phase space, our analysis is better suited for finding global features of the considered reactions. The data integrated over the longitudinal motion suggest a spectator picture of the scattering process, as it might be abstracted from the simple exchange processes, in which either one of the two slow particles, but both with the same frequency, takes over the transverse momentum of the scattered projectile. Superimposed on this pattern is an independent relative motion between the two slow particles which is characteristic of the rrN or nA system and is best observed in the direction perpendicular to the production plane. The motion perpendicular to the production plane is found to be independent of the motion in the plane. The degree of similarity (for q < 0.5 GeV/c) between the transverse momentum distributions of the pion and the baryon is however not yet naturally explained by models in which different exchange mechanisms are added, but the observed regularity suggest that such contributions are related in a def'mite way. As expected from exchange models, the longitudinal momentum fractions x of the proton fragments are not correlated with the transverse momentum transferred to the pion target (in the projectile frame) in reaction (1). However some correlation is present in reaction (2). We also tried to extend the above spectator picture to the longitudinal motion and to reconstruct the x~r distribution from that observed for py. This works quite well for large xn. We argued that the disagreement at small x n could be due to nonpomeron exchange and that we expect a dip in do/dxn at x,r "~ 0 to occur at higher energies. Comparing the nucleon and A reaction we observe almost equivalent features in the transverse momentum correlations. On the other hand, the x~r distributfons are quite different and this difference is not reproduced by the exchange models. The same holds for the correlation of ~Px ) with x=. A comparison of the x distributions in various reactions could be most suited for revealing the dynamical characteristics of the various final states after diffraction dissociation. Our analysis is limited in an essential way by the relatively low beam energy, where non-asymptotic effects are still present. A study of momentum correlations
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I¢. Ochs et aL / Proton diffraction dissociation
in experiments at Fermilab or ISR energies, as they are under way at present, can better determine whether the effects we observe are really characteristic of diffraction dissociation. In particular, it would be interesting to explore, whether or not (and if so, in which kinematic regime) our factorization properties hold at high energies. We are very grateful for enlightening discussions with G.C. Fox. We also wish to thank the SLAC bubble chamber crew, under R. Watt, and the Experimental Facilities Division, in particular L. Keller and J. Murray, for their close cooperation. In addition we thank the Data Analysis groups at our three institutions.
Appendix Here we calculate the distribution of our variables for the amplitude in (8a) (spinless particles), snn F 1/2(q 2)
(A.1)
r=g t I -m 2
In the high-energy limit we assume xP>> 1 GeV and obtain in the projectile frame sitit = 2 x i t P ( E f - qz) '
(A.2)
with the longitudinal momentum qz and energy Ef of the nf being =~ q2 qz -- 2---m '
~2aU2 Ef = (m2n + q2 + '¢z) •
(A.3)
it
Here we neglect terms of order 1 GeV2/p. Furthermore we obtain with p 2 = P2nx + P2ny' t 1 - m 2 =xit(A -P2ni 1 XitXn] '
(A.4)
where A =m 2 p
m2 it
2 mn
Xit
Xrl
The cross section is written as 3 d3pi do : ](2~') 4 ~mITI2 1-[ (2n)32E i i t t ~4(Pa +Pb - P1 - P2 - P3) i---1
(A.5)
W. Ochs et al. / Proton diffraction dissociation
427
To within our approximation, the longitudinal part of the phase space 3
q~L = 5 ( ~ E i - ~/s) 6(~,pz i - P) .i~_l(dPzi/Ei)
(A.6a)
can be evaluated as dx I t
(A.6b)
q~L ~--XnX n E f P " Finally, then
do =
1 32(2rr)4
g2F(q2 ) ( E l - qz )2 m r¢
Ef
dxndq2 dp2n±d~b
(A.7)
2 (,4 - pm/XnX n) 2 x x n '
2 ~ . where we write d2pn.t = l~ dPnld As Pnx enters the formula only quadratically we have
(Pnx) = O,
(p~rx) = - q .
(A.8)
The q dependence can be factored out (note that Ef and qz depend only on q). As a consequence the distribution in the y direction is independent of q. If we integrate over ¢) and p2ni (from 0 to oo) we obtain the factorizing form given in eq. (14). If we perform the same calculation for the nucleon exchange diagram we obtain a formula like (A.7) but with Pnz replaced by p,± and therefore, after integration, we again obtain eq. (14).
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