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Eikonal description of nucleon-nucleon polarizations
Nuclear Physics Bl17 (1976) 95-108 © North-Holland Publishing Company
EIKONAL DESCRIPTION OF NUCLEON-NUCLEON POLARIZATIONS Claude BOURRELY, Jacques SOFFER Centre de Physique Th~orique *, CNRS Marseille
Alexander MARTIN ** Service de Physique Th~orique, Centre d'Etudes Nucl~aires de Saclay, BP no. 2, 91190 Gif-sur- Yvette, France
Received 8 June 1976
Nucleon-nucleon elastic polarization data are analyzed within an eikonal framework in the range 6 ~
1. Introduction NN scattering possesses a rich spin structure which new experimental techniques are now allowing us to probe. In a previous paper [ 1] we attempted a synthesis of all available elastic proton-proton data (Plab ~> 5.5 GeV/c, - t < 4 GeV 2) within the framework of an eikonal model [2,3]. The importance of the eikonal prescription is that it allows us to unify the s and t channel aspects of two-body hadronic interactions, incorporating Regge-pole exchange with absorption and a geometrical description of the hadrons. Our exploratory attempt showed that such a description accounts successfully for the s and t dependence of the measured observables over a very large domain in energy and m o m e n t u m transfer. There were some unsatisfactory aspects in the description of the Regge-pole exchange contributions, such as the fact that the trajectories we used had no rbssemblance to the canonical secondary trajectories, and
* Postal address: 31, Chemin J. Aiguier, 13274 MarseilleCedcx 2. ** Supported in part by DRMI" Contract no. 75/1114 (CPT-Marseille). 95
96
C Bourrely et al /Nucleonnucleon polarizations
the violation of factorization. These difficulties owe their origin to certain simplifying assumptions we made, namely we had to lump together the large number of possible exchanges into one pair of effective trajectories of each signature with exchange degeneracy broken only in the coupling constants. Recent measurements of the polarization in elastic neutron-proton scattering by Diebold et al. [4] allow, for the first time, the disentangling of the isoscalar from the isovector contributions to the single flip amplitude. This adds an important element to our analysis, enabling us to overcome the difficulties in interpretation alluded to above. The result of this reanalysis of the polarization data is that we find that the isospin-one component exhibits the classic Regge behavior on output, with nearly exchange degenerate residues. In the isospin-zero channel, on the other hand, we find that we need a lower lying trajectory in conjunction with the previously postulated pomeron flip component [3]. Such a low-lying contribution is in fact, also necessary to understand the energy behavior of the data below 6 GeV/c incident momentum, although this feature of the data was not included as input to our fit. The presence of the pomeron component is indicated by the occurrence of a reversal of the sign of the polarization at high energy. While the existence of the pomeron flip coupling has been conjectured by several authors (e.g. refs. [ 5 - 7 ] ) its contribution to the polarization due to a relative phase shift between the flip and non-flip amplitudes is a natural consequence of the eikonal model. In sect. 2 we will recall the main features of the parametrization used and in sect. 3 reassess our results in the light of the new knowledge coming from tile measurement of the neutron-proton polarization. Numerical results and interpretation thereof presented in sect. 4 and some concluding remarks in sect. 5.
2. Review of the model In ref. [1] we exhibited a unified description of pp scattering from moderate accelerator energies to the highest c.m. energies available at the ISR. The vehicle for doing this was a spin-dependent eikonal model, the basic formalism and assumptions of which we wish to summarize here. The advent of ISR data with its strong diffraction minimum has confirmed the correctness of the eikonal approach. We recall that the eikonal is a complex and spin dependent function X(s, b) in terms of which the scatti~ring amplitude ~b(s, t) is expressed as ~(s, t) = ikf ( 1 -
e ix(s'
b) } eiq • b d 2 b
(1)
The eikonal itself may be interpreted as a Born term in a multiple scattering series i(l
-
e i~() :
+i
2
~__
"'"
C Bourrely et al. /Nucleon-nucleon polarizations
97
and as such represents a dynamic input to our calculation. This quantity must contain both a diffractive contribution, dominating at high energy, and the exchange of all Regge poles allowed in the t-channel. Certain simplifying assumptions are made at this point. First we assume that the non-flip amplitude is spin independent, i.e. 01 = q~3, and dominates the cross section * We work to lowest order in the helicity flip part of the Born term, as in the following
O+(s, t ) - Ol +03 2
-
k oo
f (e
1)Jo(b / )bdb,
(3)
o
Oi(s, t) = k ? ~ifs, b)ei~l(S'b)Jn(bX/'L-Db db , (4) o where the ~)i(s, b) are the Fourier Bessel transforms of the corresponding Born helicity amplitudes, n is the net s-channel helicity flip, and k the c.m. momentum. Thus for the flip amplitudes our prescription is equivalent to the standard absorption model [8]. The validity of this last assumption (eq. 4) can only by checked a posteriori in our model amplitudes. In a natural generalization of the C h o u - Y a n g [10] hypothesis we choose, for our non-flip amplitude Born term
t) = f(E
.b)F 2 (t) + R+(s, t),
(S)
where R+ is a non-asymptotic Regge-pole contribution, and the diffractive part is proportional to the convolution of the matter distribution of the two hadrons with an energy dependent coefficient of opacity f(Elab). The Regge contribution to the helicity amplitude •i is represented by 1 (C+i + C _e i - " ,*r~R(0) eaitE~a~(t)(_t)n/2 Ri(s, t) = ~-~ss
i = 1,2, 5 ,
(6)
with an(t ) = a o + a ' t and R 2 = - R 4 , since we expect unnatural parity exchanges such as n and A 1 to be negligible. Unnatural parity contributions will of course arise through the absorption mechanism, e.g. q~4(s, 0) = 0 kinematically whereas 02(s, 0) is in general non-zero. A novel feature of our work consisted in allowing the diffractive part to have spin structure with a contribution to Os given by q~diff( s'5
t) = A x / ~ f ( E Iab ) F2(t) •
(7)
This would be expected if the spin structure of the hadronic interaction were pro-
* This may be tested experimentally and is compatible with present data. Cf. ref. [9] for an appraisal of the current status of the amplitude reconstruction program.
98
C Bourrely et aL ~Nucleon-nucleon polarizations
portional to the electromagnetic current-current interaction [3], and this analogy would predict, based on the value of the isoscalar anomalous magnetic moment, a value of A = 0.06. We will show in the discussion of our results, that this sort of contribution to the eikonal leads to polarizations which do not vanish asymptotically (at least not as a power) and that there is experimental evidence for such a component in elastic 7rN and NN polarizations. A further remark about our input is in order here, and that is that, due to the many exchanges possible, namely the P', ~o, O, A= (and possibly also the 7r and e, to which we shall return later) there was no advantage to invoking factorization to reduce the number of free parameters. This model provided a coherent description of all measured pp observables, their structure in t and s-dependence over, a very large range of energy and momentum transfer, and in the next section we will show that the new pn polarization data necessitate a reexamination of the analysis of the single helicity flip amplitude Cs, alone.
3. The importance of pn polarizations An unavoidable limitation on our previous work was the absence of any data allowing us to disentangle the various Regge contributions. The lumping of Reggepole exchanges into effective even and odd signature contributions meant that no fundamental significance could be attached to the residues and trajectories we found. Significant progress toward a realistic interpretation is achieved by the extension of our analysis to include the recent measurements of the np polarizations since this allows the separation of the isospin zero and one exchanges in the single helicity flip amplitude ¢s" This may be seen roughly as follows: (i) The isovector component of the non-flip amplitude is negligible, implying (~+PP ~ gbrip, as may be verified from the fact that doPP/dt ~ danP/dt, within experimental error. An upper bound on this contribution is also given by the differential cross section for the charge exchange (CEX) process np ~ pn (pure isovector exchange) through the inequality 1 do cex 14)+(It = 1)12 ~<~" d--/-"
(8)
(ii) Since PP
(~P = ~S(It = O) +- OS(It = 1 ) ,
(9)
neglecting double flip amplitudes compared to the dominant non-flip amplitudes q)+, we can write pdo
- 2 Im(Cs¢+) =-2~bsT[,~+[
(10)
C Bourrely et al. /Nucleon-nucleon polarizations do
(11)
14+12'
d-t ~
99
and solve for the single-flip amplitude
@5(It = 0) = ~ { P ( p p )
+ P(np)),
@5(/t = 1) =
{P(pp)- P(np)),
where @ST is the component of 4s orthogonal to @+ in the complex plane, i.e. roughly the real part of 45 at small I t l. Our program is then, taking 4+ from our earlier calculations together with the new polarization data, to produce model amplitudes for the isosinglet (P, P', w, ...) and isovector components (p, Az) of @s" The phase of 4+ is important to our understanding of the polarization structure and we stress that our parametrization represents a reliable estimate for it, although in practice this quantity is unmeasurable away from the point t = 0. The reason for this is that a fixed-t dispersion relation relates the phase of an amplitude to its energy behavior. The use of a parametrization incorporating the proper analyticity requirements presumably leads to a good estimate of the phase, if the energy behavior is adequately described by the fit [11,12,13] i.e. having the proper phase is a spinoff of our fitting the cross sections over such a large range of energy. One important question of interpretation must be faced before turning to the analysis of the single flip amplitude, and that is the anomalous trajectory of the Regge-pole contribution found for @+ (real and dominant at large t). We recall that this was parametrized linearly as c~R = 0.3 + 0.5 t and, on the surface at least, cannot be identified with a degenerate P' and w trajectory, from the standpoints of extrapolating to the particle positions or of observed effective trajectories *. However, as is well known [15], this input trajectory does not correspond to the energy behavior of the eikonalized output. To estimate the latter we look at the effective power-law energy dependence of I@+RIdefined by **
~R+(s, t) = ~+(Diffractive + Regge) - ~+(Diffractive) =
f {ei(xd+xR) -
etXd]" Jo(bX/ZT) b db
o
= 7 eiXd{eiXR -- 1)Jo(bX/Z~)b db .
(12)
0
For some determinations of effective trajectories see e.g. ref. [ 14]. ** There is some ambiguity in defining the Regge contribution. This definition includes absorbed multiple Regge exchanges.
1O0
C Bourrely et al. ~Nucleon-nucleon polarizations
The effective trajectory is fairly linear in the interval 0 < - t < 1 (GeV) 2 and can be represented by a o u t p u t = 0 . 5 + 0.9t between the laboratory energies of 5 and 50 GeV/c. Since the trajectory should go through the particle positions in the unphysical region there is still a problem of principle which, however, could be resolved by ascribing some non-linearity to the input trajectory [15]. Still it is clear that we are dealing with a bona fide Regge exchange and that, at this stage, the unphysical region requirement is not germane.
4. Numerical results and interpretation The polarization data possess some interesting features which we would like to review before comparing to our results. The neutron-proton data unfortunately exist only for lab momenta of 2 to 6 GeV/c and in a rather limited range of momentum transfer, essentially It[ < 1 GeV 2. Thus, while the proton-proton data are quite extensive, the physical information contained in the sums and differences, as discussed in the previous section, is restricted to the smaller domain in which we know the neutron-proton polarization. Since our model is valid for the region above lab momentum of 5 GeV/c we can only use the new information at 6 GeV/c, which together with the proton-proton data and some plausible assumptions, will allow a reasonable extrapolation of the various contributions to higher energy and momentum transfer. Thus, for the purposes of input,to the model we are forced to neglect the very important information that the isoscalar amplitude falls faster than the isovector amplitude by approximately one power of s, though we will see later that the model solution in fact recovers this feature. At 6 GeV/c the two contributions are ap-
P
i
0,20
--
I
6 GeV/c DIEBOLD ?5 • PP
/T
L
\
OAO 0.05 0
_Q05
0.5
1.
1.5
21t I GeV2
Fig. t. Proton-proton and neutron-proton polarizations at 6 GeV/c. Data are taken from ref. [4].
C Bourrely et aL / Nucleon-nucleon polarizations
1O1
proximately equal. There is no structure in the isospin-one part while there is an indication of a double zero in the isospin-zero contribution although the data are not good enough to confirm it, much less to give its position. The pp data are generally positive with a minimum at t ~ - 0 . 9 GeV ~ at 6 GeV/c, becoming a double zero between 8 and 18 GeV/c. The latest measurements at 45 GeV/c show the polarizations becoming negative for Jtl >~ 0.5 GeV 2, within large uncertainties. This is an important effect and should be confirmed, as will be seen in the discussion below. The results of our fit are shown in figs. 1 3 with the parameters of our model listed in table 1. It will be seen that the data are reasonably well reproduced. The energy dependence of the pp polarization is correctly obtained as evident in figs. 2a, b, c which shows the comparison with the fitted data. In figs. 3a, b, c we have compared our predictions at 12.33, 14 and 24 GeV/c with available data. In all cases the t-dependence is also rather well described, even out to large t-values (I tl ~< 2.5 GeV 2) *. As a further prediction we have plotted in fig. 5 the p-p elastic polarization at 10 and 40 GeV/c. This result is obtained by the reversal of the sign of all explicitly crossing-odd terms in our parametrization and the qualitative agreement with existing data may be taken as a confirmation of the phase of our model amplitudes. The real and imaginary parts of the single-flip amplitude are shown in fig. 6. The isospin-one trajectory ressembles the one found in the non-flip amplitude and indeed we verify that this contribution has an O~eff consistent with what is known about the p-A 2 exchanges from the reactions 7r-p -+ 7r°n and n - p -+ r/n as may be seen in fig. 7. The isospin-zero component is more complicated. For the diffractive contribution the ratio of the couplings flip to non-flip is found positive and of the order of magnitude consistent with the results of the 7rN amplitudes analysis at 40 GeV/c [24]. There is now an interplay of contributions of opposite sign, the positive Regge contributions dominating at low energy with the diffractive piece winning eventually. It was not possible to obtain a fit with a single common trajectory for the isoscalar and isovector Regge exchanges and, in fact, we find the isoscalar trajectory lying roughly one unit below the isovector one. The effective trajectory corresponding to this exchange is shown in fig. 8. This contribution resembles the o (or e) trajectory invoked by Dash and Navelet [16] to explain the anomalous energy dependence of the I t = 0 ~b5 amplitude, and found also to play a role in low-energy rrN polarizations. It has been suggested by Irving [7] that a pomeron contribution to the polarizations of the sign and magnitude of the one we find, might be sufficient to explain the anomalous energy behavior, since the presence of such a term leads necessarily to a lower effective power dependence. At fixed-t, in fact, the effect becomes weaker the further one is from the energy at which tire polarization goes through zero and it is easy to check, using Irving's parametrization that this mechanism has little to do with the effect in question. * As a further check of our predictions we have made a comparison with the data out to It[ ~<5 GeV 2 (see fig. 4).
C Bourrely et al. ~Nucleon-nucleon polarizations
102
1
I
•
T
6 GeVlc • BORGHINI 70 * DIELBOLD 75
PP~PP
0.3
l
l 0.2
0.1
P!
I
1.
1.5
2
T
I
•
-1
--
__
I
I
0.5
I
2.5 I t l GeV 2 l 17. 5 GeV/c BORGHINI 71
i
0.2~ i
0.5
P ~-
.1
I
.1.5
1
:i
2.5 I t I GeV 2
2.
1
0'I ~.
I
45GeVlc GAIDOT75
_0.1 I
L
_.5
_.1
_.1.5
_.2
Itl GeV 2
Fig. 2. Proton-proton polarization data used in the fit compared to our result. The data are (a) 6 GeV/c: refs. [4,17]; (b) 17.5 GeV/c: ref. [17]; (c) 45 GeV/c: ref. [23].
C Bourrely et al. /Nucleon-nucleon polarizations
0.1
~ c /VeG 1 • ABSHIRF 2 74
103
3
-0.1 I
I
0.5
L
1.
1
_ _ _
1.5
~L_____
2.
25 It I
]7
•
,
I
PP~'PP
T-
|
OeV2
[-
7
l& GeV/c BORGHINI 71 -~
!
;
'
0.1
__
•
_
•
0.5
_
I.
•
•
1
_
L _ _ .
I
2
1.5
l
--
pp--bpp
25.
It1 GeV2 I
-
-
-I
24 GeV/c
0.05
0
-005
I
0.5
I
I.
i
I
I
I
1.5
2
2.5
Itl GeV2
Fig. 3. Proton-proton polarizations at (a) 12.33 GeM/c; (b) 14 GeM/c; (c) 24 GeV/c. Data from refs. [17,19,20].
104
C B o u r r e l y et aL ~ N u c l e o n - n u c l e o n p o l a r i z a t i o n s P
I
I
I - - T "
-I
pp_,.pp
~ -
I
I
7.9 GeV/c CERN ORSAY_OXFORD 75
0.2 0.1
+ +
j
0.
I -+1.
15
2.
25
Fig, 4. Large I t l p r o t o n - p r o t o n P
0.4 f
i
PP ---,PP
I
3.
3.5
4.
45
. Itl GeV 2
p o l a r i z a t i o n s at 7.9 G e V / c f r o m ref. [ 2 0 ] .
I
P
40 GeV/c GAIDOT 7
0.3
I
I
PP~UP
0.2
0.4 0'5I 0.3
0.I
0.2
I
10 GeV/c
0.I
i
.0.1
0
.0.2,
0.I
.0.3
0.2
-0.4-
0.3 -0.4
-D5 -0.6~_~
0.5
I
I
1.0
1.5
I t I GeM2
I
I
05
1.0
I
15 I t l GeV 2
Fig. 5. Predictions for antiproton-proton polarization compared with data at 10 GeV/c (ref. [22]) and 40 GeV/c (ref. [18]).
Table 1 It = 0
It = 1
A = 0.10 GeV - 1 s 0 = -0.44 o' = 0.88 GeV - 2 C+ = - 1 7 . 7 4 GeV - 1 C_= -2.81GeV -1 b = 1.82GeV - 2
%= 0.19 tx' = 0.45 GeV - 2 C+ = - 5 . 0 2 GeV - 1 C = 1.12GeV - 1 b = 0.138 GeV - 2
105
C. Bourrely et al. ~Nucleon-nucleon polarizations ---
i
i
6 GeV/c
_.I _.2
I ~e¢5
-.3 _./t
_.51 i
Fig. 6. The single flip helicity proton-proton amplitude ~5 at 6 GeV/c, Curves labelled I t = 0 and 1 show the isoscalar and isovector Regge contributions.
Our conclusion is, then, that the dominance of a low-lying trajectory, as found by our fit to the data above 6 GeV/c, seems to be necessary to account for the data of Diebold et al. The change of sign of the polarization at 45 GeV/c, on the other hand, indicates that the highest-lying c o n t r i b u t i o n is o f the opposite sign and it is cxeff
x/
to/
b" ~to It = 1
05
x //°//
j/x
05
1. t GeV2
;/ Fig. 7. Effective trajectory corresponding to the isovector part of ~5, computed in the energy range shown.
106
C Bourrely et aL / lVucleon-nucleon polarizations
-I'-~, ,~-°,s~
,,bZc'~,°i
s ....
I t GeV2 i t =0
j~/
q _o.s
/o/
:
//
7:
/
j
Fig. 8. Effective trajectory for the isoscalar (1t = 0, e) part of the Re~ge contribution to ~5.
P 0.2 r
I
I
0.1k ]
1- ~
pn ~ p n 6 GeV/c
0
D_N 12GeV/c
-°'I I
I 0 •2
]
'~\
%_IRVING
12GeV/c
/ [
2~ GeV/c"\ \ . ~ j
_0.3[I
0.5
J
1.
J ~
1.5 Itl GeV2
Fig. 9. Neutron-proton polarization predictions of refs. [7,16] at higher energies compared to the results our model.
C. Bourrely et al. ~Nucleon-nucleon polarizations
107
attractive to suppose it to arise from an interference of the pomeron with itself *. This effect makes essential use of the eikonal formalism since the pomeron contributions to ~+ and ~bs are of the same phase on input, and therefore do not contribute to the polarization at the level of the Born term. The polarization is generated from the absorption corrections provided by the eikonalization. It is of course hard to confirm experimentally the presence of such a constant component of the polarization. This will require precise data at higher energies especially for pn. We present the predictions of our model in fig. 9 and compare them with the predictions of the models discussed above. They are quite different and should prove useful in the interpretation of any new data.
5. Conclusion As we have seen, the eikonal model seems able to give a good description of nucleon-nucleon elastic polarizations. The input is reasonable for such a complicated set of phenomena. Thus the introduction of both the e and pomeron contributions, and the decoupling of the P', co from the single flip amplitude are suggested by the data. These results will of course be tested by measurement of the polarization at higher energy which should be forthcoming in the near future. In addition the deterruination of the isovector part of the single flip amplitude represents a significant constraint on models of the charge-exchange reaction n p - p n . Work on this direction is in progress and will be reported elsewhere. One of us (A.M.) wishes to thank R. Stora for his warm hospitality at the Centre de Physique TMorique, Marseille. We also wish to thank H. Navelet for some clarifying discussions.
References [1] C. Bourrely, A. Martin, J. Soffer and D. Wray, Argonne preprint ANL-HEP 75-41 (to be published in J. of Phys. G.). [2] R.C. Arnold, Phys. Rev. 153 (1967) 1523; A. Capella, J, Kaplan, A. Krzywicki and D. Schiff, Nuovo Cimento 63A (1969) 141. [31 C. Bourrely, J. Softer and D. Wray, Nucl. Phys. B89 (1975) 32. [4] R. Diebold et al., Phys. Rev. Letters 35 (1975) 632. [5] J. Pumplin and G.L. Kane, Phys. Rev. D I 1 (1975) 1183. * Had we identified the higher component with the f°-cv contribution we would have the ingredients of the super-Regge model [25], which does not fit the 45 GeV/c data. The contribution is there in principle and might be extracted through a detailed study of the energy dependence, as has been done in the case of rrN polarizations [26]. Since the e dominates at the energies where the isoscalar component can be extracted it is clear that one will not be able to resolve this question until pn elastic polarizations are available at higher energies.
108 [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [12] [22] [23] [24] [25] [26]
C Bourrely et al. /Nucleon-nucleon polarizations F.E. Low, Phys. Rev. D12 (1975) 163. A.C. Irving, Nucl. Phys. B101 (1975) 263. N. Sopkovich, Nuovo Cimento 26 (1962) 186. W. de Boer and J. Softer, Nucl. Instr. 136 (1976) 331. T.T. Chou and C.N. Yang, Phys. Rev. Letters 20 (1968) 1213. C. Bourrely and J. Fischer, Nucl. Phys. B61 (1973) 513. J. Bronzan, G. Kane and U. Sukhatme, Phys. Letters 49B (1974) 272. G. Hohler, H.P. Jakob and F. Kaiser, Phys. Letters 58B (1975) 348. P.D.B. Collins and A. Fitton, Nucl. Phys. B91 (1975) 332. G.L. Kane and A. Seidl, Rev. Mod. Phys. 48 (1976) 309. J. Dash and H. Navelet, Phys. Rev. D13 (1976) 1940; G. Girardi and H. Navelet, Phys. Rev. 1314 (1976) 280. M. Borghiniet al., Phys. Letters 31B (1970) 405; 36B (1971) 501. A. Gaidot et al., Phys. Letters 57B (1975) 389. G.W. Abshire et al., Phys. Rev. Letters 32 (1974) 1261. CERN-Orsay-Oxford collaboration, 1975, to be published. CERN-Orsay-Oxford collaboration, preliminary data, private communication. M. Borghini et al., Phys. Letters 36B (1971) 497. A. Gaidot et al., Phys. Letters 61B (1976) 103. J. Pierrard et al., Czech. J. Phys. B26 (1976) 13. R. Field and P. Stevens, Caltech preprint 68-543 (1975). A. Martin and H. Navelet, unpublished.
EIKONAL DESCRIPTION OF NUCLEON-NUCLEON POLARIZATIONS Claude BOURRELY, Jacques SOFFER Centre de Physique Th~orique *, CNRS Marseille
Alexander MARTIN ** Service de Physique Th~orique, Centre d'Etudes Nucl~aires de Saclay, BP no. 2, 91190 Gif-sur- Yvette, France
Received 8 June 1976
Nucleon-nucleon elastic polarization data are analyzed within an eikonal framework in the range 6 ~
1. Introduction NN scattering possesses a rich spin structure which new experimental techniques are now allowing us to probe. In a previous paper [ 1] we attempted a synthesis of all available elastic proton-proton data (Plab ~> 5.5 GeV/c, - t < 4 GeV 2) within the framework of an eikonal model [2,3]. The importance of the eikonal prescription is that it allows us to unify the s and t channel aspects of two-body hadronic interactions, incorporating Regge-pole exchange with absorption and a geometrical description of the hadrons. Our exploratory attempt showed that such a description accounts successfully for the s and t dependence of the measured observables over a very large domain in energy and m o m e n t u m transfer. There were some unsatisfactory aspects in the description of the Regge-pole exchange contributions, such as the fact that the trajectories we used had no rbssemblance to the canonical secondary trajectories, and
* Postal address: 31, Chemin J. Aiguier, 13274 MarseilleCedcx 2. ** Supported in part by DRMI" Contract no. 75/1114 (CPT-Marseille). 95
96
C Bourrely et al /Nucleonnucleon polarizations
the violation of factorization. These difficulties owe their origin to certain simplifying assumptions we made, namely we had to lump together the large number of possible exchanges into one pair of effective trajectories of each signature with exchange degeneracy broken only in the coupling constants. Recent measurements of the polarization in elastic neutron-proton scattering by Diebold et al. [4] allow, for the first time, the disentangling of the isoscalar from the isovector contributions to the single flip amplitude. This adds an important element to our analysis, enabling us to overcome the difficulties in interpretation alluded to above. The result of this reanalysis of the polarization data is that we find that the isospin-one component exhibits the classic Regge behavior on output, with nearly exchange degenerate residues. In the isospin-zero channel, on the other hand, we find that we need a lower lying trajectory in conjunction with the previously postulated pomeron flip component [3]. Such a low-lying contribution is in fact, also necessary to understand the energy behavior of the data below 6 GeV/c incident momentum, although this feature of the data was not included as input to our fit. The presence of the pomeron component is indicated by the occurrence of a reversal of the sign of the polarization at high energy. While the existence of the pomeron flip coupling has been conjectured by several authors (e.g. refs. [ 5 - 7 ] ) its contribution to the polarization due to a relative phase shift between the flip and non-flip amplitudes is a natural consequence of the eikonal model. In sect. 2 we will recall the main features of the parametrization used and in sect. 3 reassess our results in the light of the new knowledge coming from tile measurement of the neutron-proton polarization. Numerical results and interpretation thereof presented in sect. 4 and some concluding remarks in sect. 5.
2. Review of the model In ref. [1] we exhibited a unified description of pp scattering from moderate accelerator energies to the highest c.m. energies available at the ISR. The vehicle for doing this was a spin-dependent eikonal model, the basic formalism and assumptions of which we wish to summarize here. The advent of ISR data with its strong diffraction minimum has confirmed the correctness of the eikonal approach. We recall that the eikonal is a complex and spin dependent function X(s, b) in terms of which the scatti~ring amplitude ~b(s, t) is expressed as ~(s, t) = ikf ( 1 -
e ix(s'
b) } eiq • b d 2 b
(1)
The eikonal itself may be interpreted as a Born term in a multiple scattering series i(l
-
e i~() :
+i
2
~__
"'"
C Bourrely et al. /Nucleon-nucleon polarizations
97
and as such represents a dynamic input to our calculation. This quantity must contain both a diffractive contribution, dominating at high energy, and the exchange of all Regge poles allowed in the t-channel. Certain simplifying assumptions are made at this point. First we assume that the non-flip amplitude is spin independent, i.e. 01 = q~3, and dominates the cross section * We work to lowest order in the helicity flip part of the Born term, as in the following
O+(s, t ) - Ol +03 2
-
k oo
f (e
1)Jo(b / )bdb,
(3)
o
Oi(s, t) = k ? ~ifs, b)ei~l(S'b)Jn(bX/'L-Db db , (4) o where the ~)i(s, b) are the Fourier Bessel transforms of the corresponding Born helicity amplitudes, n is the net s-channel helicity flip, and k the c.m. momentum. Thus for the flip amplitudes our prescription is equivalent to the standard absorption model [8]. The validity of this last assumption (eq. 4) can only by checked a posteriori in our model amplitudes. In a natural generalization of the C h o u - Y a n g [10] hypothesis we choose, for our non-flip amplitude Born term
t) = f(E
.b)F 2 (t) + R+(s, t),
(S)
where R+ is a non-asymptotic Regge-pole contribution, and the diffractive part is proportional to the convolution of the matter distribution of the two hadrons with an energy dependent coefficient of opacity f(Elab). The Regge contribution to the helicity amplitude •i is represented by 1 (C+i + C _e i - " ,*r~R(0) eaitE~a~(t)(_t)n/2 Ri(s, t) = ~-~ss
i = 1,2, 5 ,
(6)
with an(t ) = a o + a ' t and R 2 = - R 4 , since we expect unnatural parity exchanges such as n and A 1 to be negligible. Unnatural parity contributions will of course arise through the absorption mechanism, e.g. q~4(s, 0) = 0 kinematically whereas 02(s, 0) is in general non-zero. A novel feature of our work consisted in allowing the diffractive part to have spin structure with a contribution to Os given by q~diff( s'5
t) = A x / ~ f ( E Iab ) F2(t) •
(7)
This would be expected if the spin structure of the hadronic interaction were pro-
* This may be tested experimentally and is compatible with present data. Cf. ref. [9] for an appraisal of the current status of the amplitude reconstruction program.
98
C Bourrely et aL ~Nucleon-nucleon polarizations
portional to the electromagnetic current-current interaction [3], and this analogy would predict, based on the value of the isoscalar anomalous magnetic moment, a value of A = 0.06. We will show in the discussion of our results, that this sort of contribution to the eikonal leads to polarizations which do not vanish asymptotically (at least not as a power) and that there is experimental evidence for such a component in elastic 7rN and NN polarizations. A further remark about our input is in order here, and that is that, due to the many exchanges possible, namely the P', ~o, O, A= (and possibly also the 7r and e, to which we shall return later) there was no advantage to invoking factorization to reduce the number of free parameters. This model provided a coherent description of all measured pp observables, their structure in t and s-dependence over, a very large range of energy and momentum transfer, and in the next section we will show that the new pn polarization data necessitate a reexamination of the analysis of the single helicity flip amplitude Cs, alone.
3. The importance of pn polarizations An unavoidable limitation on our previous work was the absence of any data allowing us to disentangle the various Regge contributions. The lumping of Reggepole exchanges into effective even and odd signature contributions meant that no fundamental significance could be attached to the residues and trajectories we found. Significant progress toward a realistic interpretation is achieved by the extension of our analysis to include the recent measurements of the np polarizations since this allows the separation of the isospin zero and one exchanges in the single helicity flip amplitude ¢s" This may be seen roughly as follows: (i) The isovector component of the non-flip amplitude is negligible, implying (~+PP ~ gbrip, as may be verified from the fact that doPP/dt ~ danP/dt, within experimental error. An upper bound on this contribution is also given by the differential cross section for the charge exchange (CEX) process np ~ pn (pure isovector exchange) through the inequality 1 do cex 14)+(It = 1)12 ~<~" d--/-"
(8)
(ii) Since PP
(~P = ~S(It = O) +- OS(It = 1 ) ,
(9)
neglecting double flip amplitudes compared to the dominant non-flip amplitudes q)+, we can write pdo
- 2 Im(Cs¢+) =-2~bsT[,~+[
(10)
C Bourrely et al. /Nucleon-nucleon polarizations do
(11)
14+12'
d-t ~
99
and solve for the single-flip amplitude
@5(It = 0) = ~ { P ( p p )
+ P(np)),
@5(/t = 1) =
{P(pp)- P(np)),
where @ST is the component of 4s orthogonal to @+ in the complex plane, i.e. roughly the real part of 45 at small I t l. Our program is then, taking 4+ from our earlier calculations together with the new polarization data, to produce model amplitudes for the isosinglet (P, P', w, ...) and isovector components (p, Az) of @s" The phase of 4+ is important to our understanding of the polarization structure and we stress that our parametrization represents a reliable estimate for it, although in practice this quantity is unmeasurable away from the point t = 0. The reason for this is that a fixed-t dispersion relation relates the phase of an amplitude to its energy behavior. The use of a parametrization incorporating the proper analyticity requirements presumably leads to a good estimate of the phase, if the energy behavior is adequately described by the fit [11,12,13] i.e. having the proper phase is a spinoff of our fitting the cross sections over such a large range of energy. One important question of interpretation must be faced before turning to the analysis of the single flip amplitude, and that is the anomalous trajectory of the Regge-pole contribution found for @+ (real and dominant at large t). We recall that this was parametrized linearly as c~R = 0.3 + 0.5 t and, on the surface at least, cannot be identified with a degenerate P' and w trajectory, from the standpoints of extrapolating to the particle positions or of observed effective trajectories *. However, as is well known [15], this input trajectory does not correspond to the energy behavior of the eikonalized output. To estimate the latter we look at the effective power-law energy dependence of I@+RIdefined by **
~R+(s, t) = ~+(Diffractive + Regge) - ~+(Diffractive) =
f {ei(xd+xR) -
etXd]" Jo(bX/ZT) b db
o
= 7 eiXd{eiXR -- 1)Jo(bX/Z~)b db .
(12)
0
For some determinations of effective trajectories see e.g. ref. [ 14]. ** There is some ambiguity in defining the Regge contribution. This definition includes absorbed multiple Regge exchanges.
1O0
C Bourrely et al. ~Nucleon-nucleon polarizations
The effective trajectory is fairly linear in the interval 0 < - t < 1 (GeV) 2 and can be represented by a o u t p u t = 0 . 5 + 0.9t between the laboratory energies of 5 and 50 GeV/c. Since the trajectory should go through the particle positions in the unphysical region there is still a problem of principle which, however, could be resolved by ascribing some non-linearity to the input trajectory [15]. Still it is clear that we are dealing with a bona fide Regge exchange and that, at this stage, the unphysical region requirement is not germane.
4. Numerical results and interpretation The polarization data possess some interesting features which we would like to review before comparing to our results. The neutron-proton data unfortunately exist only for lab momenta of 2 to 6 GeV/c and in a rather limited range of momentum transfer, essentially It[ < 1 GeV 2. Thus, while the proton-proton data are quite extensive, the physical information contained in the sums and differences, as discussed in the previous section, is restricted to the smaller domain in which we know the neutron-proton polarization. Since our model is valid for the region above lab momentum of 5 GeV/c we can only use the new information at 6 GeV/c, which together with the proton-proton data and some plausible assumptions, will allow a reasonable extrapolation of the various contributions to higher energy and momentum transfer. Thus, for the purposes of input,to the model we are forced to neglect the very important information that the isoscalar amplitude falls faster than the isovector amplitude by approximately one power of s, though we will see later that the model solution in fact recovers this feature. At 6 GeV/c the two contributions are ap-
P
i
0,20
--
I
6 GeV/c DIEBOLD ?5 • PP
/T
L
\
OAO 0.05 0
_Q05
0.5
1.
1.5
21t I GeV2
Fig. t. Proton-proton and neutron-proton polarizations at 6 GeV/c. Data are taken from ref. [4].
C Bourrely et aL / Nucleon-nucleon polarizations
1O1
proximately equal. There is no structure in the isospin-one part while there is an indication of a double zero in the isospin-zero contribution although the data are not good enough to confirm it, much less to give its position. The pp data are generally positive with a minimum at t ~ - 0 . 9 GeV ~ at 6 GeV/c, becoming a double zero between 8 and 18 GeV/c. The latest measurements at 45 GeV/c show the polarizations becoming negative for Jtl >~ 0.5 GeV 2, within large uncertainties. This is an important effect and should be confirmed, as will be seen in the discussion below. The results of our fit are shown in figs. 1 3 with the parameters of our model listed in table 1. It will be seen that the data are reasonably well reproduced. The energy dependence of the pp polarization is correctly obtained as evident in figs. 2a, b, c which shows the comparison with the fitted data. In figs. 3a, b, c we have compared our predictions at 12.33, 14 and 24 GeV/c with available data. In all cases the t-dependence is also rather well described, even out to large t-values (I tl ~< 2.5 GeV 2) *. As a further prediction we have plotted in fig. 5 the p-p elastic polarization at 10 and 40 GeV/c. This result is obtained by the reversal of the sign of all explicitly crossing-odd terms in our parametrization and the qualitative agreement with existing data may be taken as a confirmation of the phase of our model amplitudes. The real and imaginary parts of the single-flip amplitude are shown in fig. 6. The isospin-one trajectory ressembles the one found in the non-flip amplitude and indeed we verify that this contribution has an O~eff consistent with what is known about the p-A 2 exchanges from the reactions 7r-p -+ 7r°n and n - p -+ r/n as may be seen in fig. 7. The isospin-zero component is more complicated. For the diffractive contribution the ratio of the couplings flip to non-flip is found positive and of the order of magnitude consistent with the results of the 7rN amplitudes analysis at 40 GeV/c [24]. There is now an interplay of contributions of opposite sign, the positive Regge contributions dominating at low energy with the diffractive piece winning eventually. It was not possible to obtain a fit with a single common trajectory for the isoscalar and isovector Regge exchanges and, in fact, we find the isoscalar trajectory lying roughly one unit below the isovector one. The effective trajectory corresponding to this exchange is shown in fig. 8. This contribution resembles the o (or e) trajectory invoked by Dash and Navelet [16] to explain the anomalous energy dependence of the I t = 0 ~b5 amplitude, and found also to play a role in low-energy rrN polarizations. It has been suggested by Irving [7] that a pomeron contribution to the polarizations of the sign and magnitude of the one we find, might be sufficient to explain the anomalous energy behavior, since the presence of such a term leads necessarily to a lower effective power dependence. At fixed-t, in fact, the effect becomes weaker the further one is from the energy at which tire polarization goes through zero and it is easy to check, using Irving's parametrization that this mechanism has little to do with the effect in question. * As a further check of our predictions we have made a comparison with the data out to It[ ~<5 GeV 2 (see fig. 4).
C Bourrely et al. ~Nucleon-nucleon polarizations
102
1
I
•
T
6 GeVlc • BORGHINI 70 * DIELBOLD 75
PP~PP
0.3
l
l 0.2
0.1
P!
I
1.
1.5
2
T
I
•
-1
--
__
I
I
0.5
I
2.5 I t l GeV 2 l 17. 5 GeV/c BORGHINI 71
i
0.2~ i
0.5
P ~-
.1
I
.1.5
1
:i
2.5 I t I GeV 2
2.
1
0'I ~.
I
45GeVlc GAIDOT75
_0.1 I
L
_.5
_.1
_.1.5
_.2
Itl GeV 2
Fig. 2. Proton-proton polarization data used in the fit compared to our result. The data are (a) 6 GeV/c: refs. [4,17]; (b) 17.5 GeV/c: ref. [17]; (c) 45 GeV/c: ref. [23].
C Bourrely et al. /Nucleon-nucleon polarizations
0.1
~ c /VeG 1 • ABSHIRF 2 74
103
3
-0.1 I
I
0.5
L
1.
1
_ _ _
1.5
~L_____
2.
25 It I
]7
•
,
I
PP~'PP
T-
|
OeV2
[-
7
l& GeV/c BORGHINI 71 -~
!
;
'
0.1
__
•
_
•
0.5
_
I.
•
•
1
_
L _ _ .
I
2
1.5
l
--
pp--bpp
25.
It1 GeV2 I
-
-
-I
24 GeV/c
0.05
0
-005
I
0.5
I
I.
i
I
I
I
1.5
2
2.5
Itl GeV2
Fig. 3. Proton-proton polarizations at (a) 12.33 GeM/c; (b) 14 GeM/c; (c) 24 GeV/c. Data from refs. [17,19,20].
104
C B o u r r e l y et aL ~ N u c l e o n - n u c l e o n p o l a r i z a t i o n s P
I
I
I - - T "
-I
pp_,.pp
~ -
I
I
7.9 GeV/c CERN ORSAY_OXFORD 75
0.2 0.1
+ +
j
0.
I -+1.
15
2.
25
Fig, 4. Large I t l p r o t o n - p r o t o n P
0.4 f
i
PP ---,PP
I
3.
3.5
4.
45
. Itl GeV 2
p o l a r i z a t i o n s at 7.9 G e V / c f r o m ref. [ 2 0 ] .
I
P
40 GeV/c GAIDOT 7
0.3
I
I
PP~UP
0.2
0.4 0'5I 0.3
0.I
0.2
I
10 GeV/c
0.I
i
.0.1
0
.0.2,
0.I
.0.3
0.2
-0.4-
0.3 -0.4
-D5 -0.6~_~
0.5
I
I
1.0
1.5
I t I GeM2
I
I
05
1.0
I
15 I t l GeV 2
Fig. 5. Predictions for antiproton-proton polarization compared with data at 10 GeV/c (ref. [22]) and 40 GeV/c (ref. [18]).
Table 1 It = 0
It = 1
A = 0.10 GeV - 1 s 0 = -0.44 o' = 0.88 GeV - 2 C+ = - 1 7 . 7 4 GeV - 1 C_= -2.81GeV -1 b = 1.82GeV - 2
%= 0.19 tx' = 0.45 GeV - 2 C+ = - 5 . 0 2 GeV - 1 C = 1.12GeV - 1 b = 0.138 GeV - 2
105
C. Bourrely et al. ~Nucleon-nucleon polarizations ---
i
i
6 GeV/c
_.I _.2
I ~e¢5
-.3 _./t
_.51 i
Fig. 6. The single flip helicity proton-proton amplitude ~5 at 6 GeV/c, Curves labelled I t = 0 and 1 show the isoscalar and isovector Regge contributions.
Our conclusion is, then, that the dominance of a low-lying trajectory, as found by our fit to the data above 6 GeV/c, seems to be necessary to account for the data of Diebold et al. The change of sign of the polarization at 45 GeV/c, on the other hand, indicates that the highest-lying c o n t r i b u t i o n is o f the opposite sign and it is cxeff
x/
to/
b" ~to It = 1
05
x //°//
j/x
05
1. t GeV2
;/ Fig. 7. Effective trajectory corresponding to the isovector part of ~5, computed in the energy range shown.
106
C Bourrely et aL / lVucleon-nucleon polarizations
-I'-~, ,~-°,s~
,,bZc'~,°i
s ....
I t GeV2 i t =0
j~/
q _o.s
/o/
:
//
7:
/
j
Fig. 8. Effective trajectory for the isoscalar (1t = 0, e) part of the Re~ge contribution to ~5.
P 0.2 r
I
I
0.1k ]
1- ~
pn ~ p n 6 GeV/c
0
D_N 12GeV/c
-°'I I
I 0 •2
]
'~\
%_IRVING
12GeV/c
/ [
2~ GeV/c"\ \ . ~ j
_0.3[I
0.5
J
1.
J ~
1.5 Itl GeV2
Fig. 9. Neutron-proton polarization predictions of refs. [7,16] at higher energies compared to the results our model.
C. Bourrely et al. ~Nucleon-nucleon polarizations
107
attractive to suppose it to arise from an interference of the pomeron with itself *. This effect makes essential use of the eikonal formalism since the pomeron contributions to ~+ and ~bs are of the same phase on input, and therefore do not contribute to the polarization at the level of the Born term. The polarization is generated from the absorption corrections provided by the eikonalization. It is of course hard to confirm experimentally the presence of such a constant component of the polarization. This will require precise data at higher energies especially for pn. We present the predictions of our model in fig. 9 and compare them with the predictions of the models discussed above. They are quite different and should prove useful in the interpretation of any new data.
5. Conclusion As we have seen, the eikonal model seems able to give a good description of nucleon-nucleon elastic polarizations. The input is reasonable for such a complicated set of phenomena. Thus the introduction of both the e and pomeron contributions, and the decoupling of the P', co from the single flip amplitude are suggested by the data. These results will of course be tested by measurement of the polarization at higher energy which should be forthcoming in the near future. In addition the deterruination of the isovector part of the single flip amplitude represents a significant constraint on models of the charge-exchange reaction n p - p n . Work on this direction is in progress and will be reported elsewhere. One of us (A.M.) wishes to thank R. Stora for his warm hospitality at the Centre de Physique TMorique, Marseille. We also wish to thank H. Navelet for some clarifying discussions.
References [1] C. Bourrely, A. Martin, J. Soffer and D. Wray, Argonne preprint ANL-HEP 75-41 (to be published in J. of Phys. G.). [2] R.C. Arnold, Phys. Rev. 153 (1967) 1523; A. Capella, J, Kaplan, A. Krzywicki and D. Schiff, Nuovo Cimento 63A (1969) 141. [31 C. Bourrely, J. Softer and D. Wray, Nucl. Phys. B89 (1975) 32. [4] R. Diebold et al., Phys. Rev. Letters 35 (1975) 632. [5] J. Pumplin and G.L. Kane, Phys. Rev. D I 1 (1975) 1183. * Had we identified the higher component with the f°-cv contribution we would have the ingredients of the super-Regge model [25], which does not fit the 45 GeV/c data. The contribution is there in principle and might be extracted through a detailed study of the energy dependence, as has been done in the case of rrN polarizations [26]. Since the e dominates at the energies where the isoscalar component can be extracted it is clear that one will not be able to resolve this question until pn elastic polarizations are available at higher energies.
108 [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [12] [22] [23] [24] [25] [26]
C Bourrely et al. /Nucleon-nucleon polarizations F.E. Low, Phys. Rev. D12 (1975) 163. A.C. Irving, Nucl. Phys. B101 (1975) 263. N. Sopkovich, Nuovo Cimento 26 (1962) 186. W. de Boer and J. Softer, Nucl. Instr. 136 (1976) 331. T.T. Chou and C.N. Yang, Phys. Rev. Letters 20 (1968) 1213. C. Bourrely and J. Fischer, Nucl. Phys. B61 (1973) 513. J. Bronzan, G. Kane and U. Sukhatme, Phys. Letters 49B (1974) 272. G. Hohler, H.P. Jakob and F. Kaiser, Phys. Letters 58B (1975) 348. P.D.B. Collins and A. Fitton, Nucl. Phys. B91 (1975) 332. G.L. Kane and A. Seidl, Rev. Mod. Phys. 48 (1976) 309. J. Dash and H. Navelet, Phys. Rev. D13 (1976) 1940; G. Girardi and H. Navelet, Phys. Rev. 1314 (1976) 280. M. Borghiniet al., Phys. Letters 31B (1970) 405; 36B (1971) 501. A. Gaidot et al., Phys. Letters 57B (1975) 389. G.W. Abshire et al., Phys. Rev. Letters 32 (1974) 1261. CERN-Orsay-Oxford collaboration, 1975, to be published. CERN-Orsay-Oxford collaboration, preliminary data, private communication. M. Borghini et al., Phys. Letters 36B (1971) 497. A. Gaidot et al., Phys. Letters 61B (1976) 103. J. Pierrard et al., Czech. J. Phys. B26 (1976) 13. R. Field and P. Stevens, Caltech preprint 68-543 (1975). A. Martin and H. Navelet, unpublished.