A simple description of high energy elastic scattering

Volume 40B, number 3

PHYSICS LETTERS

A SIMPLE DESCRIPTION

OF HIGH ENERGY

10 July 1972

ELASTJC SCATTERING

G. L. K A N E *

Rutherford Laboratory, Chilton, Dideot, Berkshire, England Received 4 May 1972 Revised manuscript received 17 May 1972 Motivated by ideas about the structure of high energy amplitudes in impact parameter space, we suggest a simple form (equation six) for the dominant elastic amplitude (Pomeron exchange in Regge language). It is imaginary at t = 0 but has a significant real part away from t = 0 even at very high energies. It is able to describe the experimental features of all elastic scattering to the highest ISR energies, including the increase in slope for - t < 0.15 and considerable shrinkage in pp scattering for - t < 0.15 with little shrinkage for 0.2 < - t < 0.3; the small angle slope increase means that even elastic hadron interactions have an important peripheral contribution. The elastic amplitude has the phase variation with t needed in recent work to understand the nN scattering amplitude at 6 GeV/c in terms of the absorption model; this phase variation cannot be simply parametrized by a Regge pole. Using this Pomeron also solves the long-standing shrinkage problem for the absorption model.

For some time various workers have argued that most [1, 2] or some [3], of the features o f h a d r o n scattering amplitudes associated with quantum number exchanges could be understood in terms of simple ideas in impact parameter space. These arguments were formulated in terms o f the absorption model, or duality, and resulted in a belief in the dominance of peripheral partial waves in appropriate situations. In their simplest forms these ideas are now known to disagree with experimental data, and there is considerable controversy [2, 4] about whether they are of any significant value. In this note we extend the Use of these ideas to conjecture a simple form for the elastic amplitude (equation 6). Its significance is is threefold : (1) The amplitude we write is able to describe the experimental data [5, 6] on high energy elastic scattering in the diffractive peak, including the increase in slope [7] for - t < 0.15, and the qualitative features of the ISR data such as sizeable shrinkage for - t < 0.15 with little shrinkage for 0.2 < - t < 0.3. (Ideas in impact parameter space are not well enough understood to convincingly predict energy dependence; we cannot choose theoretically between two possibilities, one where a T = constant and the other where * J. S. Guggenheim Memorial Foundation Fellow, on Sabbatical leave from the University of Michigan.

o T increase by 5 to 10% over the region of currently possible experiments.) (2) The phase of our Pomeron amplitude as a function of t is essentially the same as that used recently in ref. [2] for rrN scattering at 6 GeV/c. Using such a phase for the elastic amplitude which goes into the absorption model, it was shown there that conventional absorption model ideas are able to account for the nN amplitudes in detail, including the phase variation of the O exchange. The Pomeron amplitude cannot be even approximately parametrized as a R e g g e pole. (3) In the past, the most difficult problem facing the absorption model [4] has been its lack of shrinkage in the region where the cuts dominate ( - t . ~ 0.2 for nonflip amplitudes). Carrying out absorption with the Pomeron o f equation 6 also solves this problem [ 16], giving good agreement with the shrinkage of n - p -+ n°n at least out to - t = 1.5 GeV 2 . The effect arises in a simple way, with the edge contributions giving absorptive cuts which decrease in size and shrink as the energy increases. In ref. [2] it was found that it was possible to describe 6 GeV/c nN amplitudes and data with the absorption model, if one used an elastic amplitude with a certain (reasonable but not independently determined) phase. It was observed there that it was possible to interpret the elastic phase as arising partially 363

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from f exchange and partially from a Pomeron contribution with a real part described by Re M ~ exp(B! t)R'x/ZTJ1 ( R ' x / ~ ) ,

(1)

with R ' a radius somewhat less than one fm. The argument to obtain the form ofeq. (1) was as follows. In addition to an important imaginary central contribution, the Pomeron can have (though it is not compulsory) a real part. We expect that real part to arise in much the same manner as typical non-elastic hadron interactions, with an important contribution from the edge in impact parameter space (because it is not determined by the coherent shadow of all the inelastic processes). We also want to require that at t = 0 the Pomeron is purely imaginary. Then there must be another contribution in addition to that from the edge near one fro; presumably this contribution is from a shorter range. If these contributions were delta functions in impact parameters we would obtain ReM which was the difference of two J0's at different radii. When the edge is spread out [8] we obtain these with multiplicative smooth functions such as exponentials. Very crudely, we can replace the difference of two J0's by the derivative times the difference of arguments at the two radii, so we arrive at eq. (1). B 1 gives the amount of spreading of the edge; usually B 1 < 2 GeV -2. If this interpretation of ReM at 6 GeV/c is correct, then we are also led to have an edge contribution to ImM, of the form exp (B 0 t)Jo(R x/-t) for the nonflip amplitude. In addition to the intuitive appeal of such a contribution, it is well known that if one has an analytic amplitude whose real part (in impact parameter, or the J-plane) has a zero, then the imaginary part will be sizeable and will have a peak nearby (i.e. presumably at the edge). Assuming the dominant imaginary central contribution can be described simply by A exp(Bt), we then put lm M ~ A exp (Bt) + A 0 exp(Bot)Jo(R x / ~ ) .

(2)

If we only needed to study t-dependence this form would be adequate. If, in addition, we want to consider s-dependence (or analyticity) we must be more careful. To study s-dependence in terms of impact parameter ideas [8] is very hard and very few definite 364

10 July 1972

results have emerged. If, however, we think in terms of the absorption model for Reggeon exchange we recall that J0 appeared in order to simply reproduce the systematics of an absorbed Regge pole. Thus we assume an equivalence

Jo(Rn/Zt) ,, Pole-Cut, where "Cut" represents [1] the absorption correction which generally behaves as a cut in the angular momentum plane. If the pole exhibites shrinkage one has at t=0 Cut = constant/In s.

(4)

Altogether then, motivated both by these ideas and by the data, we conjecture that R 2 = R 2 Ins,

R '2 = R02 lns,

(5)

and

M(s, t) = is [A exp(Bt) + A 0 exp(B0 t) J0(R x/-~)] + sA 1exp(B1 t) R ' x / ~ J1 (R'x/--i),

(6)

or possibly

A 0 exp(Bot)Jo(Rx/~) ~, Ao(exp(Bot)lns) -(G/lns) exp(B0t In ½ s).

(7)

Except for details of energy dependence or analyticity, eq. (6) is the useful form to think of for our elastic amplitude. The main difference between (6) and (7) is that in the latter case one has o T rising a few percent to its constant limit (since the term with 1/lns is destructive and goes away) while with (6) one has o y = constant. When the experimental situation is settled then one form can be ignored. Although all the quantities (except overall scale) in equations ( 5 ) - ( 7 ) are in principle fairly well known, in practice our understanding of impact parameter ideas is not good enough to completely fix them. We know that R must be about one f m ; B 0 and B 1 smaller than or about 2(GeV/c) -2 ; B characteristic of scattering from a disc of radius of order of one fro; and the total contributions from the edge and whole

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Table 1 Values used in calculating the curves shown in figs. 1- 3. The first column gives the results for eq. (6), shown in figs. 1 and 3. The second and third give results for the substitution of eq. (7), I for the upper curves and II for the lower curves in fig. 2.

Jo A A0 A1 B Bo

55 40 13 3 2

Bz

2

R0 Rb G

2.1 1.7

Pole -Cut I

II

53 71 23 5.4 2 2

78 33 15 4.5 1.75 2

2

2

1.25

1.0

disc in a ratio characteristic of the area of the edge compared to the area of the whole disc, less than or of order 1. In eq. (7) we know that the Bessel function and the absorptive form have the same zero as a function of t at --t ~ 0.2, the same value at t = 0 at some s, and similar shrinkage. With all these constraints in mind, we have determined a set of values for these "constrained parameters" by fitting a mixture of selected real data and hypothetical data chosen to insure that the results were similar to the preliminary ISR data; they are given in table 1 and are all reasonable. Figure 1 shows a typical result using the J0 in eq. (6). A few data points are shown from the experiment of Edelstein et al. [9] at PL = 29.7 GeV/c (S = 60 GeV2). At the higher (ISR) energies the preliminary slopes reported in ref. [6] are shown in parentheses along with the slopes from the model, for 0.05 < - t < 0.1, and for 0.2 < - t < 0.25. The model appears to describe the situation very well. The total cross section is constant at 37.3 rob, consistent with the preliminary quoted value from ref. [6] of 37.5 -+ 1.5 mb. The noticable shrinkage for - t < 0 . 1 5 (due to the shrinkage of J 0 ( R 0 ~ ins)), and the essential absence of shrinkage for - t < 0.2 (due to the importance of the real part in that region) are a natural consequence of the model and are clearly apparent in the data of ref. [6]. To show the effect of a rising o T we show in fig. 2 two typical sets of curves, again with slopes, and

10 July 1972

100 = 37. 3 mb (37. 5 ,,1.5)] ~t(7(pp --pp)

103(1~'-02 I0.5(10.8~:

£ ~I~

S=2808

0.05

Q!

0.2 -t (C~Wc)2

0.3

Fig. 1. The differential cross sections for pp elastic scattering at three high energies, arising from equation 6, are shown. For comparison, typical data are shown at S = 60 GeV2 (PL = 30 GeV/c) and the slopes and constant oT from the preliminary ISR data of ref. [6] are shown in parentheses along with the values for the model. The shrinkage at small t and the lack of shrinkage at larger t are characteristic of the model and are present in the data. with typical data from ref. [9]. Although the slopes are generally steeper, these curves are not inconsistant with the preliminary data of ref. [6] and the fact that they fall below the S = 60 GeV 2 data could be attributed to leaving out the contributions of secondary Reggeons there (such as co and f). The p h e n o m e n o n of shrinkage at small t and little shrinkage at larger t is again present (for the same reasons as above) but is a little more complicated now. Essentially all of the increase to a higher value of do/dt (and thus aT) at t = 0 occurs in the last 0.05 (GeV/c) 2 in t, so extrapolations of that distance could be misleading if o T is rising. Finally, in fig. 3 we show the real and imaginary parts of the pp elastic amplitude as function of t at two energies, and for comparison we show the results 365

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10 July 1972

1.0

~_~(PP~PP}

Pomeron Amplitude ImM

'~'~%~

~ ) d'rtrr~)

<.I

.05<-t

+.,\ I--12,0. I 4+.2 -:'...-..n-

,~ lOO

+

(Xcrt (PP]/{~t(nP) )

?c

5{ 3(

o,t

26

01

10 8 6 5 t~

.....

x

I-.-12808 .05

I 4ts I 0.1

.15

0.2

.25

0.3

.35

- t (GeV~)

Fig. 2. The results are shown for the case of total cross sections rising to a constant value, with the substitution of eq. (7). The oT value at each energy is given in parentheses at t = 0. The value reached by ¢rT at infinite energy is in the range of 50-55 mb. Some slopes are shown. from the 6 GeV/c ?rN analysis of ref. [2], renormalized by the ratio of pp and p total cross sections (these +rN results for the " P o m e r o n " rather than for the full isoscalar exchange amplitude at 6 GeV/c are obtained by subtracting a "reasonable" f exchange contribution and so are mainly of qualitative significance). For - t > 0.3 the real part can play an important role even in the cross section (not to speak of the polarisation). A number of remarks should be made at this stage. (1) The single form in eq. (6) with Bessel functions is meant to be a useful memonic and should (presumably) be treated sceptically. In particular, when studying crossing or analyticity properties, in any variable, or signature properties, or details of the energy dependence, the alternative forms in eq. (7) (for example) give different answers. E.g. one can have pole

"~.

/ /

02

"~

I

1~

"..

-0.11" ~

slope (GeV"=1

3

366

S= 2808 GeV 2 S =60 S = 121TN

O.&

0.6 0.8 1.0 - t ( GeVle ) 2

1.2

l.&

1.6

Fig. 3. The real and imaginary parts of the pp amplitude are shown for the results of fig. 1 (other cases are not very different), and compared with the results for +rN scattering from ref. [2]. minus cut in the J-plane or complex moving/-plane branch points [8, 10]. Similarly, the real part R'X/ZTJ1 (R ' ~ 2 T ) is really a difference of two J0's one stage earlier and a difference of two ( p o l e - c u t ) terms one more stage earlier. None of these properties will show up in the data, but they should be kept in mind to avoid unnecessary theoretical confusion. It is amusing that the J0 form may represent data better than the absorption forms which led to it. One possibility is that it is representing a situation [15] where the entire cut discontinuity in the J-plane vanishes at t = 0 so that the l / I n s term is decoupled there; this does not occur for conventional Reggeon absorption cuts, but the situation could be different for the Pomeron. (2) We are aware of two other possibly related attempts to describe the same features of the very high energy elastic data. One, by Edelstein [11 ] , interprets the result in terms of mixing between the proton and 1 • J = ~- diffraction dissociation states; it is possible that this approach is connected to ours and accounts in the

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imaginary part for some of the origin of the Bessel function terms*. Second, Barger and Phillips [12] have fitted the data with no real part at any t and a J0 in the imaginary part which is only important at small t. The interpretation in impact parameter space of their amplitude is thus very different from the interpretation of ours. Some comments about other approaches are given by C. Michael in ref. [4]. (3) To study n, K, ~ processes one only needs to reinterpret the co-efficients. Again, our understanding of impact parameter ideas is not good enough to confidently predict what should happen. If the relative importance of the edge effects to the entire disc go down as o T decreases, we would expect the increase in slope to occur at smaller t**. If one multiplies the disc term by oT0rp)/ow(pp ) and the edge term by that squared, one finds for the J0 case a 7rN slope of 8.3 GeV - 2 at 30 GeV/c, to be compared with 8.4 + 0.2 at 25 and 40 GeV/c**.To do a complete and consistent treatment of ~p and pp reactions or K-+p reactions, as is necessary, we must include the secondary Reggeon contributions from f and co even at Serpukhov energies. There does not appear to be any difficulty that will arise in doing this, particularly when the results concerning the energy dependence of crT at the ISR are confirmed. (4) Interpretation of elastic polarisations in all reactions will be strongly affected if the Pomeron amplitude has the phase we give it. It has already been shown in ref. [2] that the double zero in 7r+-p elastic polarisations at - t ~ 0.7 can be interpreted in a new way with a large part of the effect arising from interference of the real part of the Pomeron amplitude and the imaginary part of the p flip amplitude. Similarly, it appears that the double zero in the pp elastic polarisation near - t = 1, which is not understandable in terms of the usual duality picture with an imaginary Pomeron and a real Regge amplitude, arises here partially from the increase of the real part after its zero. * The possibility that the diffraction dissociation reactions also arise mainly from the edge and can be described simply in terms of impact parameter ideas is discussed by CohenTannoudji et al. [13]. ** This is what is observed by the CERN-ITEP Collaboration [ 14] ; they find the optical point lies above the extrapolated do/dt at 25 GeV/c and 40 GeV/c but actual breaks are not observed in the slopes.

10 July 1972

(5) An amusing point can be noted if it is verified that one should have two radii R and R ' < R to describe Medge. Then one will find some t values where the zeros of J0(R x/rL~) and J1 (R'x/'Z~-) will coincide. In particular, if at low energies we have R ~ 5 and R'/R = 0.8 then Medge would vanish at - t ~ 3(GeV/c)2, which could account for the fixed t dip noted near that point in elastic processes. (6) Attempts to separate the Pomeron from secondary exchanges such as f or co will be complicated by the presence of the real part since then one cannot factor out a purely imaginary dominant Pomeron. (7) Secondary high lying Reggeons such as f and co must be included in a complete analysis of the elastic data. They can fully account for the real part of the amplitude at t = 0, and they may have a significant effect on do/dt away from t = 0 at PL ~ 30 GeV/c. When the ISR data is fully available it will be possible to make an analysis which determines the dominant amplitude at the highest energies and then go down in energy to include the Reggeons at lower energies. (8) If our description remains a valid one it will be difficult to untangle the apparently dominant s-channel unitarity and absorptive effects to find the underlying structure of a t-channel exchange. We have written a simple elastic amplitude which (i) can be motivated using simple ideas in impact parameter space, (ii) allows one to solve the main problems of the absorption model in describing twobody reactions, and (iii) allows one to describe the ISR data on pp elastic scattering, including the increase in slope for small t and the shrinkage properties. On the basis of these results one is presumably justified in having considerably increased confidence in the validity of the point of view we have used. Unfortunately, it will not be easy to verify directly that the Pomeron has an important peripheral contribution or that it has an important real part; it may be that we can become convinced of the answer only in terms of the internal consistency of increasingly general models. I would like to thank M. Block, R. Worden and G. Ross for useful suggestions and particularly M. Vaughn for helpful discussions and help with the comparison with data. 367

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References [ 1 ] M. Ross, F. Henyey and G. L. Kane, Nucl. Phys. B23 (1970) 269. [2] B. J. Hartley and G. L. Kane, Phys. Letters 39B (1972) 531. [3] H. Harari, Phys. Rev. Letters 26 (1971) 1400. [4] For some arguments on these topics see R. J. N. Phillips, Amsterdam Conf. Proc.; C. Michael, invited talk Fourth Intern. Conf. on High energy collisions, Oxford, April 1972; V. Barger, K. Geer and F. Halzen, CERN preprint TH.1470, to be published. [5] B. Diddens, Fourth Intern. Conf. on High energy collisions, Oxford, April 1972.

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[6] G. Barbiellini et al., Fourth Intern. Conf. on High energy collisions, Oxford, April 1972, presented by P. Strolin. [7] R. Carrigan, Phys. Rev. Letters 24 (1970) 168. [8] H. Harari and A. Schwimmer, Phys. Rev., to be published. [9] R. Edelstein et al., Carnegie-Mellon Univ., preprint to be published. [10] B. Desai, Phys. Rev. Letters (1972). [ 11 ] R. Edelstein, Fourth Intern. Conf. on High energy collisions, Oxford, April 1972. [12] V. Barger and R. J. N. Phillips, to be published. [13] G. Cohen-Tannoudji, G. L. Kane and C. Quigg, Nucl. Phys. B. (1972). [ 14] M. N. Kienzle, Fourth Intern. Conf. on High energy collisions, Oxford, April 1972. [15] H. Abarbanel, NAL preprint, to be published. [16] B. J. Hartley and G. L. Kane, work in progress.