The shape of the quasi-elastic peak

Nuclear Physics B82 (1974) 189-200. North-Holland Publishing Company

THE SHAPE OF THE QUASI-ELASTIC PEAK M. JACOB and R. STROYNOWSKI

CERN, Geneva Received 10 July 1974 Abstract: The analysis of single diffractive excitation at very high energies suggests that the corresponding differential cross section could be very different from that found fo* elastic diffractive scattering. Differential cross section and polarization measurements for Itl < 1 (GeV/c) 2, could show prominent structures, the existence of which is motivated. This is particularly the case if one assumes that the reaction occurs predominantly without helicity flip as expected for a highly coherent process.

1. Differential cross section in diffractive excitation Among the important recent results of research at ISR energies, one should certainly quote evidence for the prominent role of diffractive excitation in particle production [ 1]. Such a mechanism has a particularly clear signature in the case of single diffractive excitation, whereby one proton is quasi-elastically scattered while the other one flares into a hadronic state which eventually results in a many-particle system. Such a mechanism is characterized by a low-momentum transfer, which makes it an a priori highly coherent process. It is also characterized by no quantum number exchange, as inferred from the analysis of the reaction in terms of an exchange process, a picture which is justified by the first mentioned property. It has finally an energy dependence which, for each specific final state, is relatively weak or in any case comporable to that of the elastic cross section. These three key features actually define what is meant by diffractive excitation. They were known to apply to the production of specific hadronic resonances clearly seen at PS energy [2]. They are now also known to apply to a very broad spectrum of excited states which altogether involve a single diffractive cross section similar to, if slightly smaller than the elastic cross section. Its mass distribution extends up to 10 GeV at least [1]. The diffractive excitation cross section do/dM 2 appears to decrease with the excitation m a s s M asM - 2 at large M [1,3]. In any case, converging evidence from ISR and from NAL support the existence of a sizeable cross section for the single diffractive excitation of relatively h e a w hadronic states [1,4]. Analyzing such a reaction as a quasi-two-body process, one may define a quasi-elastic differential cross section in terms of the m o m e n t u m transfer distribution of the isolated proton

190

M. Jacob, R. Stroynowski, The shape of the quasi-elastic peak

recoiling from the excited one. Information about this distribution is still fragmentary. In the 1 0 - 3 0 GeV range of machine energy much variety was reported according to the type of excited state actually observed. The distribution could be very narrow(as compared to that observed for elastic scattering), as in the case of the N*(1470) where a slope of 18.3 is given, or much broader, though narrow still, as in the case of the N + ,(1968) where one quotes a slope of 7.4 [2]. The slope is generally said to become broader with increasing missing mass [5 ]. The precise shape in the neighbourhood of t = 0 is still the object of discussion and it may show a dip at t -- 0. Nevertheless, most of the cross section is actually confined to the low It[ region. Indeed, coherent production on nuclei, which is sharply cut-off in Ltl by the nucleus form factor, is known to be quite strong [6]. Diffractive excitation involves mainly low momentum transfers, a property which does not conflict with its tentative association with a highly coherent inelastic process. At much higher energies (ISR and NAL) mass resolution problems are such that one has to consider rather wide excitation mass ranges as a whole (ZXM~ 2 GeV, say). Track chamber data [4] and counter data * at NAL would then support a relatively steep slope, thus approximating the differential cross section as e -81tl at low It[, while spectrometer data at ISR [1], relevant to large ItL values (LtL > 0.4, say), would give smaller slope, approximating the differential cross section as e -41tl , say. Fig. 1 puts together most of the data avilable at present *. When globally perused, they suggest a behaviour which is very different from elastic scattering. Certainly better experimental studies are needed. At present, the experimental data are given with rather large errors and relatively poor mass resolution. They do not indicate any striking structure and show rather little energy dependence. It is indeed interesting to contrast this global shape with the behaviour of the elastic cross section at ISR energies. The latter shows a striking diffractive structure with a sharp minimum at Itl = 1.4 (GeV/e) 2 [8]. This dip occurs after a fall by over 6 orders of magnitude from the maximum value at t = 0. If parametrized in terms of an eponential at low Itl, the slope is rather large, increasing from 11 to 12.5, say, as one spans the ISR energy range [8]. Elastic scattering and quasi-elastic scattering, even if both diffractive in nature, seem to show rather different structures in t. It is argued in the following that this should be expected. This makes it the more interesting to study the corresponding distributions over the full t range through a single experiment as probably possible at the ISR and certainly at the SPS (or NAL).

2. The proton shape and opacity This section, which defines notation and recalls some relevant relations, is writ* The data summarized in fig. 1 are to be found in the refs. [7],

M. Jacob, R. Stroynowski, The shape of the quasi-elastic peak

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Fig. t. Differential cross section distribution for the reaction p p - , p+ X. Data are those listed in ref. [7]. They are separated in two groups, those corresponding to M 2 ~ 10 GeV 2 and those corresponding to M 2 ~ 40 GeV 2. The latter value corresponds to the limit for the kinematical separation of the diffractive component at NAL energies. ten in a review style for the sake o f c o m p l e t e n e s s . One m a y first n e g l e c t a n y h e l i c i t y flip a m p l i t u d e in elastic s c a t t e r i n g a n d write:

do

M(s, t)[ 2

dt-

16rrs 2

(1)

l m A ( s , 0) = s Oto t.

(2)

with

A is the invariant scattering amplitude, s and t are the centre-of-mass energy squared and the four-momentum transfer squared, respectively. It is then standard procedure to introduce a partial-wave expansion in terms of the impact parameter r as a(r, s) = ~

I f A ( < _q2) e _ i q . r J 8rrs d2q'

or, c o n v e r s e l y ,

(3)

M. Jacob, R. Stroynowski, The shape of the quasi-elastic peak

192

A(s, t) - f a(r, S)Jo(rx/-t)rdr. 8ns

(4)

The centre-of-mass three-momentum transfer is denoted by q (q2 = t). The approximations and relations involved in writing (3) or (4) are standard and well known. It is a resonable approximation to assume that, at ISR energies, the elastic scattering amplitude has become practically pure imaginary, at least in the rather low [tj range where the cross section is sizeable *. The inelasticity parameter r7 is then simply related to the scattering amplitude a as

~(r, s) = 1 - 2 la(r, s)l ,

(5)

with

A(s, t) = 4isw/~ I( d°~ ~ dt ] "

(6)

The ratio of the inelastic cross section to the elastic cross section at fixed impact parameter is equal to Oin(r )

1-[r/(r)l 2

1-1a(r)l

%l(r)

[l-r/(r)[ 2

[a(r)[

'

(7)

Oin = 8 7 r f la(r)l(1-la(r)[) r dr.

(s)

with, as a result

One may thus describe the proton shape in terms of the impact parameter amplitude

a(r) or in terms of the inelastic cross section at fixed impact parameter. The latter description corresponds to the overlap function approach [10] ~. The set of equations recalled above makes the corresponding procedure straightforward in the pure inraginary no spin-flip approximation. The picture simplifies further if one expands do/dt (1) in terms of a sum of exponentials in t (or Gaussians in Iq[). This is, in particular, the approach followed by Jackson in ref. [1 1]. The Bessel transform of a Gaussian being a Gaussian, the amplitude: * Measurements o f the ratio of the real to imaginary part of the elastic forward amplitude give a relatively small value which actually vanishes through the NAL energy range [9]. The mere observation o f a dip at [tl = 1.4 (GeV/c) 2, where the imaginary part most probably vanishes, further stands for a relatively small real part t h r o u g h o u t the low ltl range. :~ Imapct parameter analysis of elastic-scattering data has been carried in detail in several recent papers [11].

M Jacob, R. Stroynowski, The shape of the quasi-elastic peak A(t) = is

~oe i e }bit

193

(9)

l

corresponding to the differential cross section dc~ dt

16rrl /~/.. oti~fet2(bi+bf)t

(10)

implies the partial-wave distribution:

a(r) = L_ 8rr i °ei

e- r2 / 2bi (11)

bi

The radius associated with an exponential distribution is r i = x/2b i. A "typical" slope of 10(GeV/c) -2 corresponds to a radius of 0.9 fm. In terms ofa(r) such a rough description of the differential cross section gives a Gaussian distribution in the impact parameter. The inelastic cross section (or the related opacity) then varies as

°in(r)-

1 d°in - ° e- re/2b ( 1 - - -°- e

27rr dr

27r

b

8rrb

rZ/2b }

'

(12)

which gives a flatter distribution in the neighbourhood of r = 0 than that ofa(r), though falling eventually the same way at large r. The unitarity limit on the righthand side is 1, while o/2rrb(l- (o/8rrb)) is empirically of the order of 0.9. The proton looks rather dark at the centre, but the opacity falls quickly as one moves away from r = 0. All this is by now well known [1 1 ]. This has been quickly reviewed here for the sake of completeness, introducing at the same time paranaeters and relations of importance for the following discussion. Reproducing the structure seen in elastic scattering requires little change from this picture. The simplest approximation now involves two exponentials with opposite signs in (9) and, as a result, two Gaussians with opposite signs in (11). To the extent that the second maximum is down by almost six orders of magnitude as compared to the forward peak, this means but a very small depletion from a purely Gaussian shape in the neighbourhood of t = 0.

3. Impact parameter picture for diffractive excitation The shape of the proton, as obtained from (12) is that most likely expected from a rather soft object 1 fm across, as soon as one assumes that, at such high energies, semi-classical (or optical) arguments should be reliable. Going further, one may try to calculate separately different contributions to (12), assuming that the pertinent processes would show but little coherence among themselves. One may thus distinguish "show-case" diffraction with medium missing masses (1 < M < 6 GeV,

M. Jacob, R. Stroynowski, The shape o f the quasi-elastic peak

194

say) from an a priori complicated remainder for which one may also a priori expect the Gaussian-like structure of (12). This is, however, not the case if the quasielastic peak is similar to the elastic one. If it were the case, and if an average slope B could be considered as providing a reasonable approximation, one would get a specific contribution to the (imaginary) amplitude, which -- in analogy with (11) would read OLD e r2 / 2B aD(r) - 87T B

(13)

M2/s is assumed to be small enough so that the Fourier transform can be performed as done in the case of elastic scattering. The corresponding cross section is

4 ½°D - 167rB

(14)

the factor { on the left-hand side corresponds to our summing over both proton diffractions in order to get o D. The normalized diffractive amplitude for either of the two (incoherent) diffractive processes is dD(r ) =]/70-7 e - r 2/2t? ? 8rrB

(15)

and the total inelastic contribution to Oin(r ) in (12) reads

-~

(16)

It has, embarrassingly, a maximum at r = 0 while the success of absorption models at providing a reasonable description of quasi-two-body reactions would suggest a rather peripheral process. Furthermore, fig. 2a displays the r dependence of ain(r) and e.D (r) ' respectively, together with their difference. Following our apIn proximation of purely imaginary amplitudes, this corresponds to the contribution of all other, i.e., non-diffractive inelastic, processes. When we would a priori expect a flatted Gaussian distribution, we find that the complex remainder would give more opaqueness on the edge of the proton than in the centre. This point was already noted long ago by Michejda [12] who did not find it as embarrassing as suggested here. It should be stressed though that quite a sizeable cross section is now associated with single diffractive excitation. The resulting effect becomes then conspicuous. This was indeed recently emphasized as a difficulty by Sakai and White [13]. They rather propose important helicity-flip contributions. They advocate for them, on the ground that helicity could be conversed in the t-channel and not in the s-channel, as often suggested. The relations of the preceding section are easily generalized to an arbitrary helicity flip AX. Instead of (4) one writes:

M. Jacob, R. Stroynowski, The shape of the quasi-elasticpeak

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2

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Fig. 2. Inealstic cross section profile as a function of impact parameter r. The total inelastic cross section is calculated from eq. (12), using the data o f the CERN-Rome group at x/s = 53 GeV. The diffractive distribution on fig. 2a is calculated from e~l. (16 with o D = 6 mb and a slope parameter B = 8 (GeV/c) ÷2. The diffractive cross section on fig. 2b is calculated from eq. (21), as described in the text.

A ,ax 8~s

= faAx(r, s),d,~x(rq) r dr,

(17)

with conversely AAX a a x ( r , s) = ~F8-~-sJax(rq)

qdq.

A h e l i c i t y a m p l i t u d e w i t h n e t flip AX v a n i s h e s at least as qAX at q = ~

(is) = 0. We

M. Jacob, R. Stroynowski, The shape of the quasi-elastic peak

196

may write out such a factor explicitly and get the partial waves corresponding to a Gaussian-type distribution

A zxx = Ao(qx/b) ax e- ~q2b , as

A 0 r/'Xb-I r2/2b a a x ( r , s ) - 8ns (2tAX+ 1) e

(19)

which has then an obvious peripheral shape. S u m m i n g again over b o t h protons, the corresponding cross section is

- --X2

b-I

(20)

OAX - 87rs2

The profile corresponding to (18) corresponds more to what is e x p e c t e d from a diffractive excitation amplitude than (15) does *.

4. Can zero helicity flip still d o m i n a t e ? There is, however, no reason why even a zero helicity flip amplitude should behave as (15). One may easily introduce the required structure writing instead d o ( r ) = o~(~B)~ e-r 2/2B

[j(~)½

e-r2/2B ' "

(21)

This corresponds to the most simple guess, with parametrization as in the case o f elastic scattering [11]. We choose B' < B and a//3 in such a way that the corresponding profile of oDn(r ) n o w has a m i n i m u m at r = 0 together with a peripheral shape. The p r o d u c t i o n amplitude may formally be l o o k e d at as the result of the * Our arguments cannot be definitive as they now stand. Nevertheless, we expect that, first diffractive excitation should not distinguish itself very much from all other quasi-two-body processes and, second that a strongly peripheral structure for an a priori complicated remainder is conversely unlikely enough that it is worth exploring in detail the pertinent consequences. Furthermore, associating a peripheral structure to a multiperipheral production process, on the grounds of an exponential cut-off in t at each step has been proposed but it does not survive relaxing strong assumptions about phases. This was shown long ago by Michejda in ref. [ 12]. Finally in view of the fact that, as discussed in ref. [111, the increase of the total cross section is mainly of a peripheral nature and also of the fact that it could be due to diffractive excitation, as recently discussed by Amati and Caneschi [ 14] the peripheral nature of diffractive excitation is worth probing further. Our present description of diffractive amplitudes was briefly presented by one of us (M.J.) [ 15] in the CERN Academic Training Programme (1974).

M. Jacob, R. Stroynowski, The shape of the quasi-elasticpeak

197

destructive interference between a long-range component B and a shorter-range component B' in analogy again with what one envisages for high-energy elastic scattering. Fig. 2b gives the corresponding profile for OiDn(r) and the remainder Oin(r)--oD(r). It is obtained using (16) with the following choice of parameters B = 8 GeV - 2 ,

B' = 4 GeV

2,

O~20c~ ~2o~

= 2,

o o = 6 mb.

This is rather the situation which we would like to advocate as worth checking for. We keep a dominant zero helicity flip amplitude, as usually considered for elastic scattering *, but impose what we consider the proper structure in impact parameter. Considering diffractive excitation as a coherent process we also note that the semi-classical condition for helicity flips to set in ~ / - t >1 l/R, where R is the range, is also that for coherence to disappear. The profile of the remainder, which has now a Gaussian-like shape at low r and eventually falls off smoothly, is also quite compatible with what expected from the over-all absorption provided by the many other reaction channels. Taking (21) as a diffractive amplitude, we can calculate the corresponding differential cross section from (10) and (11). The destructive interference needed in order to obtain a dip at r = 0 in the impact parameter profile, now results in a very pronounced dip in the neighbourhood of It[ = 0.3 (GeV/c) 2. This is shown in fig. 3. This very prominent structure, as opposed to what is observed in elastic scattering, most generally results of course from the fact that one has to impose interference between amplitudes of a similar order of magnitude. To say the least, the data summarized in fig. 1 do not strongly support it**. This is, however, not a good reason to give it up and one should rather see how sensitive the structure is to our introducing extra effects so far neglected in the straightforward calculation presented here. To that end we have considered three effects which all could, if strong enough, provide a smearing of the dip into what * We do not wish to rule out helicity flip umplitudes. Nevertheless, the fact that diffractive cross sections off nuclei are quite important, when the nucleus form factors could appreciably quench helicity flip contributions leads us to explore in detail how the zero helicity flip amplitude could be dominant, Alignment measurements of diffractively produced hadronic states would be very interesting at NAL (SPS) energies. They are also needed in order to see if anything happens at the no-flip component as t goes to zero. From a semiclassical picture, the one unit flip amplitude should not be appreciably quenched kinematically beyond Itl = 0.05 (GeV/c) 2. However, one loses coherence as helicity flip may set in. Such a coherence requirement could be stronger for the diffractive background than for the prominent resonances. **There is however so much averaging made in the data that more refined measurements could show the looked-for dip.

M. Jacob, R. Stroynowski, The shape of the quasi-elastic peak

198

104

,

~

I

I I I04~-~~

I I I

cos 0 : 0 , 8 J

cos8 : 0 9

I0:

10 3

qO2 ~5 <~

b

i(

I0

16

o

04

08

t,2

i6

Fig. 3. Differential cross section distribution for the zero helicity flip amplitude with destructive interference as described in the text and with helicity flip admixture (3a) or phase difference between the larger and smaller Ltl contributions (3b). The relative amount of helicity flip cross section is denoted by 6. The phase difference between the two terms considerd is denoted by 0. is seen in fig. I. The first one consists in letting the slope parameter B (and B') vary with M. As already m e n t i o n e d , there is evidence that it does and, as a result, dips for different values of M will n o t be located at the same value of t when the data o f fig. 1 imply a sizeable averaging. We, however, c a n n o t consider too strong a variation and it is f o u n d that such an effect would by itself be quite unsatisfactory at reconciling fig. 1 and fig. 3. Narrowing the missing mass range would of course help*. The n e x t possibility consists of i n t r o d u c i n g helicity flip c o n t r i b u t i o n s . All needed relations have been given. The different curves given in fig. 3a correspond to the int r o d u c t i o n of helicity flip c o n t r i b u t i o n s kept, however, at a relatively l o w level**. Our conclusion is that our somewhat c u m b e r s o m e structure in t is only weakly sensitive to our departing from an overwhelming zero helicity reaction. This may be also inferred from the results of Sakai and White [ 13] which require very large helicity flip c o n t r i b u t i o n s (t-channel helicity conservation in their approach) in order to reproduce the data. This does n o t rule out the presence o f a sizeable helicity flip and the detailed study of the diffractive differential cross section (at fixed M 2) * As would limiting oneself to exclusive channels. ** We take the same B dependence for the leading non-flip and the full single flip amplitudes. The latter one is already peripheral as such. We include helicity flip one only. Introducing higher flip components does not alter our argument about the dip at Itl = 0.3. It would, of course, enhance the resulting dip at t = 0.

34. Jacob, R. Stroynowski, The shape of the quasi-elastic peak

199

in the neighbourhood of t = 0 at NAL (SPS) energies appears as of prominent importance *. Nevertheless, we cannot consider a small admixture of helicity flip as enough to solve our embarrassing discrepancy between fig. I and fig. 3. The last effect not yet considered is to depart from a purely imaginary amplitude. The dip is then associated with the vanishing o f the imaginary part. The real part, which has most generally a different t-dependence than the imaginary part, should not vanish (if it ever does) for the same value of t. There are, however, a priori many ways to parametrize such an effect. We have chosen a very simple one according to which the different t-dependence of the imaginary and real part is globally translated in terms of a phase difference between the two components of (21). We do not advocate that it should in general be done this way but our point is rather that a relatively small phase difference is enough at smearing out the dip. This is shown in fig. 3b, where one sees the curves which correspond to different but all relatively small phase differences. This effect is, of course, related to the fact that the contributions of the two terms included in our approximation should not be very different in magnitude. As a result, a small phase change has a big effect on their required destructive interference. One can therefore, in principle, reconcile the data of fig. l with a dominant peripheral zero flip production if one departs from a purely imaginary amplitude. The conclusion is now that the presence o f such a phase, even i f a relatively small one, together with a priori relatively small but also present flip amplitudes, should result in polarization effects which should be particularly observable in the region of the expected zero in the imaginary part of the zero flip amplitude, namely around ttl = 0.3 (GeV/c) 2, and where the cross section is still large enough that the corresponding measurement could be feasible 2. It therefore appears as worth while both to analyze in greater detail the t dependence o f the quasi-elastic peak, which is of prominent importance at NAL (SPS) energies, and to search for polarization effects in the corresponding missing mass experiments.

References [1] M.G. Albrow et al., Phys. Letters 42B (1972) 279; Nucl. Phys. B51 (1973) 388; B54 (1973) 6; M. Jacob, CERN-Dubna School 1973, Ebeltoft, CERN Yellow report (1973); CERN Academic Training Programme, Lecture Notes, CERN Yellow report (1974); Important results from the CERN-Holland-Lancaster-Manchester Collaboration and from the Aachen-CERN-Genova-Harvard-Torino Collaboration received timely reviews in ISR meeting summary papers, CERN Internal reports (1973 and 1974). * This is also very interesting in view of the possible vanishing of the pomeron coupling at zero momentum transfer squared, as discussed for instance in [16]. :[:We are grateful to L. Dick for a discussion of polarization measurements in inelastic collisions. New developments in polarized targets should make such measurements possible. We also thank A. Kotanski for discussions.

200

M. Jacob, R. Stroynowski, The shape of the quasi-elastic peak

[2] F. Zachaxiasen, Phys. Reports 2 (1971) 1; R.M. Edelstein et al., Phys. Rev. D5 (1972) 1073; R. Slansky, Phys. Reports 11 (1974) 99. [3] A.C. Melissinos, Invited Paper APS Chicago Meeting (1974); S, Childress et al., Phys. Rev. Letters 32 (1974) 389; Proc. Vanderbilt Conf., AIP Conf. Proc. No. 12 (1973). [4] J. Whitmore, Phys. Reports 10 (1974) 273; Michigan-Rochester Track Chamber Collaboration at NAL, preprint (1974). [5] D.R.O. Morrison, Rapporteur's Talk at the Kiev Conf., 1970; S. Humble, CERN preprint TH. 1872 (1974); H.I. Miettinen, Moriond Conf., 1974. [6] W. Beusch, Lecture at 12th Cracow School of Theoretical Physics, Zakopane, 1972, Acta Phys. Pol. B3 (1972) 679. [7] HBC 205 GeV/c: J.S. Barish, D.C. Colley, P.F. Schultz and J. Whitmore, Phys. Rev. Letters 31 (1973) 1080; ISR l I + 11 GeV/c: M.G. Albrow et ah, Nuch Phys. B., to be published; ISR 15 + 15 GeV/c: M.G. Albrow et ah, Nucl. Phys. B54 (1973) 6; NAL 300 GeV/c: S. Childress, P. Franzini, J. Lee-Franzini, R. McCarthy and R.D. Schamberger, Phys. Rev. Letters 32 (1974) 389; NAL 200 GeV/c: S. Childress, P. Franzini, J. Eee-Franzini, R. McCarthy and R.D. Schamberger, Proc. Vanderbilt Conf., 1973. [8] U. Amaldi, Rapporteur's talk, Proc. Aix-en-Provence Conf., 1973. [9] US-Soviet "gas-jet" Collaboration at NAL, prepfint (1974). [10] L. Van Hove, Rev. Mod. Phys. 36 (1964)655. [ 11 ] R. Henzi and P. Valin, McGill preprints (1973); F. Henyey et al., Michigan preprint (1973); H. Miettinen and P. Piril~, private communication. The subject is reviewed by: J.D. Jackson, Scottish Universities Summer School (1973); U. Amaldi, ref. [8]. [12] L. Michejda, Forthschr. Phys. 16 (1968) 707; L. Michejda, T. Tumau and A. Biat-as, Nuovo Cimento 56A (1968) 241. [13] J.N.J. White, GIFT publication 7 (1973); N. Sakai and J.N.J. White, Nucl. Phys. B59 (1973) 5 l l ; See H.I. Miettinen, ref. [51 for a detailed discussion of such effects. [14] D. Amati and L. Caneschi, Proc. of Aix-en-Provence Conf., 1973. [ 15 ] M. Jacob, CERN Academic Training Programme, 1974, CERN Yellow report 7 4 - 1 5 (ref. [ 1 ]). [ 16] Ya.I. Azimov et al., Proc. of the 1 l t h Winter School on nuclear and elementary particle physics, Leningrad 2 (1974) 5.