Nuclear Physics B73 (1974) 429-439. North-Holland Publishing Company
ELASTIC SCATTERING
FROM THE TWO-COMPONENT PICTURE
F.S. HENYEY* Max-Planck-Institut fit'r Physik und Astrophysik Munich, Germany Received 30 November 1973 Abstract: The momentum transfer dependence of high-energy elastic scattering is discussed in the context of the multiperipheral + diffraction picture. Both a "naive" and an "absorbed" model are considered. Fits are made to the impact parameter dependence of the pp elastic amplitude. It is essential in this picture that the diffractive component be peripheral; in the fits it has a zero at Itl between 0.2 and 0.4. The origin of the small t-break in this picture is discussed, al;d contrasted with its explanation in similar fits.
1. Introduction The two-component picture [ 1 ] for multiparticle production is being widely used to describe various aspects of multiparticle production data. In such a picture, multiparticle production is imagined to occur by either of two processes. The first is diffraction, while the second is usually taken to be multiperipheral (or at least similar to multiperipheral). It is interesting to test such a picture by comparing it not only to multiparticle data, but also to elastic scattering data by calculating the predicted elastic scattering using s-channel unitarity. Such a calculation is conveniently done in the impact parameter representation [2]. The diffractive component is usually assumed (as we shall assume in this paper) to be a quasi two body process, and its contribution to elastic scattering is given by the familiar form Odiff(b) = ~
13~rdiff(b)12,
(1)
up to a factor depending on the norm~ization chosen for the amplitude. The sum is taken over all diffraction channels. Qdiff is called the overlap function [3] of diffractive scattering. In the "naive" two-component picture, the imaginary part of elastic scattering is given in terms of the overlap functions by Imh~el(b) = Ill'el(b)12 + Odiff + O'MPM " * On leave from the University of Michigan, Ann Arbor, Mich., US.
(2t
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430
The multiperipheral contribution, O M P M , does not have as familiar a form as the diffractive contribution. Recently the multiperipheral contribution has been investigated in the impact parameter representation [4, 5]. The results of these investigations are described in sect. 2. In this paper, the multiperipheral component is a simple multiperipheral cluster model, as in ref. [4]. The momentum transfers are approximated by their transverse parts and the sunenergies by functions of rapidity differences. Such things as low subenergy correlation effects, interference between particles produced at different points in the multiperipheral chain, and resonance formation are lumped together into an effect called "clustering". The extra slowing of the leading particles at high multiplicity caused by energy conservation is neglected. The "naive" two-component picture treats diffraction as entirely unrelated to the multiperipheral component, except that, in this paper, diffraction is constrained to be no larger than the multiperipheral component at any impact parameter. On the other hand, in this paper the diffractive component is closely related to the elas-
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F i g . 1. The impact parameter and m o m e n t u m transfer dependences of the "naive" model. The " d a t a " curve is taken from ref. [13]. The decomposition into the components is shown. The diffractive component alone as a function of b is shown in fig. 3. The zero in the diffractive component near t = - 0 . 4 is related to its peripheratity.
F.S. Henyey, Two-component picture
431
Table 1 Parameter values of the fits Parameter
"Naive" model
Absorbed model
G
24.9 mb
58.4 mb
A
16 mb (fixed)
16 mb (fixed)
R2o
1.725 GeV -2
2.00 GeV -2
h-+ 2
4.00 (not allowed to be smaller)
2.36
R]
4.79 GeV -2
6.07 GeV -2 (Odiff(b = 0) = 0)
The normalization chosen here and in the figures is ato t = ImM(0) = i f Imll~(b) db 2 . In each case A was fixed and one further parameter was at the end of an allowed range.
tic component in the sense that JMell2 + Odiff is assumed to have a simple parameterization. It is, of course, somewhat incorrect to take a dominantly imaginary amplitude (i.e. diffraction) to be unrelated to other scattering, since its imaginary part is related via unitarity to other scattering. A result of the neglect of this constraint is that Regge cuts have a positive sign rather than the correct negative sign [6, 7]. These difficulties are partially remedied in the other model considered in this paper, the absorbed multiperipheral model [8]. In this model, the multiperipheral component is reduced by absorption. The particular model ofref. [8] considers only absorption by elastic scattering. It is known however, [ 9 - 1 1 ], both theoretically and phenomenologically, that diffraction dissociation also provides absorption. This point has been recently demonstrated by Blankenbecler [ 12], who shows a reversal in sign of the contributions of imaginary amplitudes to the total cross section. The absorption correction is a factor - 2 times the elastic and diffractive components. This leads to the consideration, as an alternative to eq. (2), of the model Imll~el(b) = OMPM - Odiff - JJl~el(b) 12"
(3)
The same parametrizations will be used for this "absorptive" model as for the "naive" model. The usual practice in making a comparison with elastic data is to transform the model amplitude into the momentum transfer representation, and compare its square with the elastic differential cross section. A preferable practice was initiated recently [13], in which the comparison is carried out in the impact parameter representation. Phase and spin assumptions are made about elastic scattering, (Essentially it is assumed that both the real part and helicity flip contribute little to the cross section) and the elastic scattering amplitude is transformed to impact parameter, where it is compared to the model overlap function.
432
F.S. Henyey, Two-componentpicture
For a complete theory, these two approaches would be equivalent. However, present models contain essentially arbitrary functions, whose form is usually chosen for economy of parametrization. It then becomes important to distinguish between structure which is intrinsic in the model, and structure due to details. Such a distinction is made, in the case of s-channel dynamics, by looking in the impact parameter representation. In high-energy pp elastic scattering it appears that the most important structure is the large t-dip, and that the small t-break is a minor structure. Such a view might be entirely reasonable within the context of t-channel dynamics. However, in the impact parameter representation, exactly the opposite case is seen to occur [13]. The large t-dip is extremely subtle, being equivalent only to a 10% change in slope of the amplitude at small impact parameters. Such a small effect could easily be changed by a choice of parametrization. On the other hand, the small t-break is equivalent to a surprisingly long range part of the amplitude. Its range, ( b 2 ), assuming a reasonable extrapolation to small b, is three times the (b 2 ) of all elastic scattering. Beyond 2 fm, this "tail" is dominant. The existence of this tail is not a subtle effect in the context of s-channel dynamics, and should be explainable by a multiparticle model. In the remainder of the paper, the multiperipheral and diffractive components are discussed, and the fits to the data are presented. Comparison is made to related work [14-16].
2. Multiperipheral component In this section the results of ref. [4] are reviewed (see also ref. [5]), and the multiperipheral component is described. The multiperipheral amplitude corresponds to a random walk in impact parameter space. At each step of the walk a cluster of particles is emitted. The rapidity is most negative at the beginning of the walk and increases until it is most positive at the end of the walk. As a rough approximation, which is used in this paper, the leading clusters carry off all the incident energy. As a result of angular momentum conservation the overall impact parameter is just the distance between these clusters in this approximation. It is to be assumed that the random walk is entirely unconstrianed; any actual constraint is to be included in the definition of the cluster size. With no clustering, the elastic slope is much too steep, so clustering must be included. If, as expected, the random walk step size is energy independent, the cluster size must grow rapidly with energy between, for example, 100 and 1500 GeV in order that the observed rapid increase in multiplicity be consistent with the observed slow shrinkage of elastic scattering. If the cluster size is increasing fast, a In s fit to the average multiplicity might be meaningless in the contect of the multiperipheral model. The increase in cluster size might be due to a number of possible mechanisms.
F.S. Henyey, Two-component picture
433
The increase of the total cross section may be due to the threshold for production of higher mass resonances (e.g. slightly above NN threshold) which have a decay mode into many pions [ 17]. The increase of observed multiplicity would also be caused by the production of these resonances. It might be that the asymptotic average cluster size is very large [5] (or even infinite), so that finite energy effect limit the cluster size. This is apparently the case in the CLA model [18]. For a very large cluster size the elastic slope can be obtained by making the uncertainty principle relating the step size to the average transverse momentum of the cluster be far from an equzlity. It may be that the first and last steps are large, but the intermediate ones are small [19], essentially putting all pions in one cluster. This possibility, however, seems to lead to a disagreement between the experimental avarage transverse momentum and the minimum allowed by the abovementioned uncertainty principle. In what follows, the energy dependence question is ignored, and the cluster size is taken to be an adjustable parameter at each energy. The probability of producing n + 2 clusters in the multiperipheral model is (approximately) a Poisson distribution in n. Thus the random walk of n + 1 steps is weighted with the Poisson probability e-~n-n/n!. Thus, i f f (b) is the step size distribution, the multiperipheral overlap funtion is given by QMPM = G ~ e - n ~ n / n ! (T(b)*) n+l
(4)
t/
The notation ( ~ ' , ) n + l means the n + 1 fold convolution of 7 with itself. The transform of this to momentum space is particularly simple, since the transform of a convolution is a product
~n OMPM(t ) = G ~-J e - ~ n i f ( t ) n+ l n
= G f ( t ) e x p [ ~ ( f ( t ) - 1)],
(5)
where f ( t ) is the transform ofT(b) and is normalized so that f(O) = 1. The fits presented later in the paper are based on e ~ . (4) and (5). The adjustable parameters are G, ~, and the simple function f ( t ) = eRo t, with one parameter,R0, is used. With this choice o f f ( t ) , one immediately sees that O (t) is positive at all t-values. The total overlap function however does have a zero at t = - 0.6, as first pointed out by Zachariasen [20]. The result of the absence of such a zero in the multiperipheral c o m p o n e n t s almost independent of the form o f f ( t ) , f ( t ) cannot get very negative, becausef(b) is positive, being the absolute square of a random walk step amplitude. Rigorously, f ( t ) > - 0.403; however it is unreasonable to expect f ( t ) to come close to attaining this bound. Moreover, i f ~ is moderately large, the exponential in eq. (5) prevents OMpM(t) from becoming very negative. This arises by a cancellation between terms of even multiplicity, for w h i c h f ( t ) n+l is negative and
F.S. Henyey, Two-componentpicture
434
terms of odd multiplicity for which it is positive. Thus the negative value of the overlap function at [tl > 0.6 indicates something other than the multiperipheral component, in particular to the diffractive component and/or absorptive effect, to which we now turn.
3. Diffractive component and absorptive affects In this section the peripherality of diffraction dissociation is reviewed, evidence for this peripherality from our considerations is deduced, and a simple parametrization is chosen. The peripheral nature of diffraction dissociation has been argued from various points of view. Ross et al. [ 11 ] suggested that strong cut absorption phenomenology, especially for pion exchange reactions, would be improved by including diffraction dissociation of a longer range than elastic scattering. This peripherality was attributed to absorption of the diffraction dissociation amplitude. This suggest-
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F.S. Henyey, Two-component picture
435
tion was utilized by Hartley and Kane [21] as a major ingredient in their quantitative fit to many (quantum number exchange) reactions. Various direct phenomenological considerations argue for the peripherality of diffraction dissociation. For example, peripherality is necessary to explain the steep slope of low mass diffraction dissociation [22]. Sakai and White [15, 23] derive peripherality form inclusive data and approximate t-channel helicity conservation. Their mechanism is equivalent to attributing the peripherality to the presence of large (s-channel) helicity flip amplitude, as well as to absorption. These large helicity flip amplitude also occur in the work of Kane [22], to account for the slope-mass correlation. The considerations of the present paper aJso require the peripherality of diffraction dissociation. This is most easily seen separately for the naive and absorbed models. In the naive model, the zero of the overlap function at t = - 0.6 requires this peripherality. Since the multiperipheral component is expected to remain positive, the zero must be caused by a cancellation with the diffractive component. Thus the diffraction dissociation component of the overlap function must have q zero between t = 0 and } = - 0.6. This statement is direct evidence for peripherality; a disc of radius one fermi corresponds to a zero at t = - 0.6, while a one fermi ring corresponds to a zero at t = - 0.2. In the absorbed model, the zero in the overlap function is explained by absorption. Absorption effects naturally lead to a zero at - 0.6, since the hadronic size is about 1 fm. In this model the peripherality is related to the fact that the "tail" in impact parameter begins only around 2 fm, while the elastic scattering drops off after about 1 fm. The diffraction dissociation must therefore play a role between one and two fermis, yet not be too large below one fermi so that its integrated size is reasonable. A simple parametrization of a peripheral function is as a difference of two exponentials. Since diffraction dissociation is so closely related to elastic scattering, and since the peripherality may be at least partially caused by absorption, it is reasonable to take the smaller exponential as closely approximating the elastic contribution to its own imaginary part. This leads to the adoption of the form ;7 ~diff + 1~'1el 12
2R12
e
,
(6)
to be used in either eq. ~ ) (naive) or (3) (absorbed). The two parameters A and R 1 are constrianed so that Oaiff is positive at all b. The transform of eq. (6) to momentum transfer is 2
Odi ff + M * * M = A e R ~t
4. Fit to data
The considerations of the previous sections have led to the models
(7)
F.S. Henyey, Two-component picture
436
(8)
ImMel(t ) = G e x p [ R 2 t +-~(e R2ot - 1)1 +-Ae R ~ t ,
where the -+ signs refer to the naive and absorbed models respectively. Fits were done at one energy only, (S = 2800 p-p scattering), as the energy dependence question is not resolved. In both fits, the parameter A was fixed at 16 rob. This is arrived at by 7 mb elastic +7 mb single dissociation (estimate of many experimentalists and phenomenologists) +2 mb double dissociation (factorization). The parameter G then is fit by the total cross section. The remaining parameters were then fit by a comparison to the shape of the experimental data, both in t and in b. A fit to the large b tail was emphasized. In each model one further parameter took a value at the end of a range to which it was restricted. In the naive model, the average number of clusters ~ was not allowed to become smaller than 2. In the absorbed
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IMPACT PARAMETER Fig. 3. The peripheral diffractive components of the two fits, compared with the curve of Sakai and White, who used entirely separate considerations from those of this paper. The absorbed model has a more peripheral diffractive component than the naive model.
F.S. Henyey, Two-component picture
437
model this constraint was not able to be imposed without destroying the fit. A perhaps unreasonably small value for F was obtained in this model: however absorption preferentially removes low multiplicities. Inclusion of pion exchange in f of eq. (5) would be expected to allow these values o f f to become larger, as explained below. Finally, in the absorbed model A/2R 2 was f'Lxed at the value which makes Odiff(b = 0) =0. The parameters of the fits are shown in the table, and the fits are shown in the figures. Is can be seen that the multiperipheral component smoothly changes slope. The steep slope at small inpact parameter corresponds to low multiplicities; the random walk cannot get far. In the absorbed model thege low multiplicities are supressed. Likewise, the shallow slope at large impact parameter, which gives the large b tail, corresponds to a random walk of many steps. Thus the multiperipheral model provides a natural explanation of the large b tail. This explanantion is related to that of ref. [13]. In ref. [13] it was proposed that the large b tail is generated by whatever happens at small impact parameter plus one or more exchanges. The multiperipheral model provides a special case of this mechanism, the small impact parameter part also is generated by a series of exchanges. The small values of the average numbers of clusters in the fits occur in order to make the transition from the central region to the tail occur sufficiently rapidly. Because of its long range nature, pion exchange would be expected to cause this rapid transition, allowing the average number of clusters to be larger. Thus the fitted values o f f should not be taken seriously. It should be noted that the fit would not be as good without an approximation that was made, namely that the overall impact parameter is the distance between the beginning and end of the random walk. In fact [2, 4], it is somewhat reduced for high multiplicities. The large cluster size of our results provides some justification for the approximation, but this feature may lead to some difficulty with detailed models of the multiperipheral component. The diffractive component, in either model, makes the change in slope near two fermis more abrupt than the multiperipheral model alone. In order to do this, the diffractive component is peripheral, as shown in fig. 3, compared to the calculation of Sakai and White based on entirely separate considerations. Barshay [16], De Groot and Miettinen [14], and Amaldi [24] have associated the break at small t with the peripheral nature of diffraction. This small t break is closely associated with the large b tail. Thus the mechanism presented here differs markedly from theirs. An important distiction needs to be made between the moderately peripheral region 1 fm < b < 2 fro, in which peripheral diffraction is important in the picture presented in this paper, and the very peripheral region b < 2 fm, which is associated with the break in t, and which is unrealted to the diffractive component. In addition to the fact that there exists an alternative mechanism from multiperipheral dynamics, or any other danymics which does not forbid exchanges, there is a good reason that the large b tail cannot be diffractive. Diffraction dissoci-
438
F.S. Henyey, Two-component picture
ation is the elastic-like transition between two similar but not identical states. In order to have the coherence of the imaginary parts, the scattering of these two states into relevant multiparticle state connot be extremely different. As a result, diffractive scattering can only occur at any given impact parameter when there is a. significant amount of non-diffractive scattering. In particular, any reasonable model of diffraction dissociation never allows more diffractive than non-diffractive scattering. Thus the large b tail should not be more than 50% diffractive. Moreover, the sharp change in slope around 2 fm. suggests that the scattering beyond this point is almost entirely non-diffractive. The present paper is, in a sense, complementary to that of Sakai and White [15]. They were not specifically concerned with the small t-break, since they did not consider impact parameters of 2-3 fm. They made the separation into diffractive plus non-diffractive by concentrating on the diffractive component, whereas the present paper concentrates on the non-diffractive component, assuming it to be multiperipheral. It can be seen, from fig. 3, that these two view-points are in agreement, in th',k the separation is nearly the same.
5. Summary (i) The multiperipheral plus or minus diffraction two-component pictures are able to fit the t and b dependence of elastic scattering moderately well. (ii) The large b tail is made from high multiplicity multiperipheral processes. This is easily understood as a long random walk in impact parameter. (iii) The fit requires a small number of large clusters. This conclusion might be altered by including pion exchange. (iv) In either of the two models it is required that diffraction be peripheral. This result is in agreement with other theoretical and phenomenological considerations. Diffraction, however, is not periphreal enough to account for the small t-break. (v) The energy dependence question remains open. The number of clusters grows much more slowly than the number of produced particles. I would like to thank U. Amaldi for a discussion which led to this work, to A. Kryzwicki for the hospitality of the Laboratoire de Physique Theorique et Particules Elementaires, Orsay, where this work was started, to the theory group et the MaxPlanck-Institute for their hospitality, and to N. Sakai for discussions and for reading the manuscript.
References [1] C. Quigg and J.D. Jackson, NAL preprint THY-93 (1972); W. Ftazer, R. Peccei, S. Pinsky and C. Tan, Phys. Rev. D7 (1973) 2647;
F,S. Henyey, Two-component picture
[2] [3] [4] [5] [6] [7] [8] [9] 10] 11] [12] 13] 14] 15] 116] 17] 18] 19] 20] [21] [22] [23] [24]
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H. Harari and E. Rabinovici, Phys. Letters 43B (1973) 49; K. Fialkowski and H.I. Miettinen, Phys. Letters 43B (1973) 61; L. Van Hove,Phys. Letters 43B (1973) 65. F.S. Henyey,Phys. Letters 45B (1973) 363. L. Van Hove, Rev. Mod. Phys. 36 (1964) 665; Nuovo Cimento 28 (1963) 798. F.S. Henyey, Phys. Letters 45 B (1973) 469. C.J. Hamer and R.F. Peierls, Phys. Rev. D8 (1973) 1358. V.A. Abramovskii et al., Proc. of the 16th Int. Conf. on high-energy physics, Batavia (1972) p. 389. R. Blankenbecler et al., SLAC@UB 1381, to be published, and references therein. D. Amati et al., CERN preprint TH. 1676, to be published. V.N. Gribov et al., Phys. Rev. 139 (1965) B184. F.S. Henyey et al., Phys. Rev. 182 (1969) 1579 M. Ross et al., Nucl. Phys. B23 (1970) 269. R. Blankenbecler, Phys. Rev. Letters 31 (1973) 964. F.S. Henyey, R. Hong Tuan and G.L. Kane, Michigan preprint UM-HE-73-18, to be published. E.H. De Groot and H.I. Miettinen, The pomeron ed., J. Tran Thanh Van (Orsay, 1973). N. Sakai and J.N.J. White, Nucl. Phys. B59 (1973) 511. S. Barshay, Nuovo Cimento Letters 3 (1972) 369. G. Chew, private communication. H.M. Chan et al., Nuovo Cimento A57 (1968) 93. C. Chiu, Texas preprint (1973). F. Zachariasen, Phys. Reports 2 (1971) 1. B.J. Hartley and G.L. Kane, Nucl. Phys. B57 (1973) 157. G.L. Kane, Acta Phys. Pol. B3 (1973) 845. N. Sakai and J.N.J. White, Max-Planck-Institute preprint (June 1973). U. Amaldi, Talk at 2nd Aix-en-Provence Int. Conf. on elementary particles (1973).