Regge cuts and πN charge-exchange polarisation

Nuclear Physics B33 (1971) 614-620. North-Holland Publishing Company

R E G G E C U T S A N D rrN C H A R G E - E X C H A N G E

POLARISATION

J.W.COLEMAN* and R.C.JOHNSON Mathematics Department, University of Durham, England

Received 28 June 1971

Abstract: The process rr-p ~ 7tOn at high energy is described by exchange of the O-pole plus o ~ P cut. Models with and without nonsense zeros are considered, the latter finding it easier to explain new polarisation measurements at 5 and 8 GeV/c.

1. INTRODUCTION New measurements [1 ] show large values of polarisation 9 in 7rN charge-exchange scattering at 5 and 8 GeV/c, indicating the presence of significant corrections to pure p-meson Regge-pole exchange. Here we investigate this process at small angles ( - t < l (GeV/c) 2) in terms of the exchange of the p-pole plus the p ~ P Regge cut. Pole amplitudes both with and without nonsense zeros are considered, and for the cut contribution we introduce a simple ansatz which has features suggested by the absorptive/eikonal (AE) prescriptions [2,3], including in particular relatively isotropic t-dependence and destructive sign. The models are fitted to cross-section data subject to some finite-energy sum rule (FESR) constraints, and the polarisation is predicted. Although a conclusive distinction is not possible, the model without nonsense zeros, which has strong cut contributions, finds it much easier to explain the new data.

2. POLES AND CUTS (i) Pole. The p-meson Regge-pole contributions to the usual amplitudes [4] A ' and vB are written R r = iNTr exp (art)(-iv)~ ,

* SRC research student.

(1)

J. W. Coleman and R. C.Johnson, Regge cuts

615

( r = A , B ) , where v = ~ ( s - u ) and the trajectory a is approximated by a ( t ) = 1 + ( t - m 2 ) a '. These terms are explicitly crossing-odd, with the correct asymptotic phase

given by Re R r / I m R r = tan 17ra. The residues have exponential t-dependence extracted, the factor N equals either 1 (Model M - no nonsense zeros), or a(t) (Model A - with nonsense zeros), and the remaining couplings 3'r are assumed constant. (ii) C u t . The contributions from the p ~ P Regge cut are represented by the simple expression C r = iX r exp [arbt/(ar+b)] ( - i v ) ac [c+ln (-iv)] ~ .

(2)

These amplitudes have the correct crossing properties, and so the proper asymptotic phase given by Re C r / I m C r = tan ½7r0, where 0 = a c + 2~ arctan

7r

7r 2 [c+ln v~ "

(3)

The branch-point trajectory is a c = 1 - m 2 a ' + t a ' / ( 2 a ' + 1), corresponding to an effective Pomeron trajectory of 1 + 0.5t, and the parameter/3 is constrained by tchannel unitarity [5] to be less than - 1 . The AE models in fact have/3 = - 1 (refs. [2,3] ); perhaps/3 + 1 is small. The parameter c allows for the possibility of a different scale of energy-dependence in the logarithmic part of the cut, (e.g. i f c is large then the logarithmic factor varies relatively more slowly). It is included because such a term appears in the AE cuts [2,3], where its value is determined by (in our notation) ar, b and a'. We assume here that both c and/3 are effectively constants for Itl < 1 (GeV/c) 2, and also that they are independent of amplitude A' or B - that is, the energy dependence of the cut (like that of the pole) is a t-channel J-plane feature. The form of the exponential t-dependence in eq. (2) is based explicitly on the AE models [2,3]. The parameter a r is the p-pole residue slope parameter, (as in eq. (1)), and b is the corresponding quantity for the Pomeron. High-energy 7rN elastic scattering data [6] give b ~ 4 (GeV/c) -2. Possible factors in eq. (2) which are polynomials in t, arising through the detailed AE mechanisms from e.g. the nonsense factors in the p-pole, are refinements which are relatively unimportant for Itl < 1 (GeV/c) 2, (they do not affect the phase of the amplitudes). The sign of the coupling constants ~kr is chosen opposite to that of the pole, 7r. This not only is given by the AE prescriptions [2,3], but also is supported by considerable phenomenological and theoretical evidence, (see e.g. Lovelace [7] for a review). Thus the cut amplitude of eq. (2) is constructed consistent with known requirements, and constitutes a destructive and relatively isotropic correction to the pole term, having the effect of absorption in the low partial waves. Its strength, which

J. W.Coleman and R. C.Johnson, Regge cuts

616

we propose to determine phenomenologically is expected to be larger in non-flip (A') than in flip (B).

3. MODELS AND FITS

(i) Model A. We consider first a model where the pole amplitudes have nonsense zeros, i.e. N = a(t) in eq. (1). There is good support for this kind of model [7,8], which gives the following picture of the charge-exchange process: (a) the dip-bump structure of do/dt comes from the nonsense zero in the Ocoupling to the dominating spin-flip amplitude B; (b) there is a negligible cut term in B, but a substantial contribution in the nonflip amplitude A', where the nonsense zero is moved off the real axis to give a crossover zero in Im A ' near t = - 0 . 2 (GeV/c) 2, (c) the dominant term in the polarisation is proportional to Im (R~CA), which contains a factor a sin ~n(a-O), and which therefore has one zero at a = 0 and othersata=O,O + 2 , 0 + 4 , ...etc. We note that the structure of the amplitudes as implied by points (a) and (b) is generally consistent with the preferences of FESR's [9], which tend to suggest a simple approximate relation between Re B and Im B (e.g. a pole phase factor tan 17ra) but a much less straightforward link between Re A ' and Im A', (see especially fig. 9 of ref. [9] ). The dip mechanism determines a' = 0.86 GeV -2 and XB = 0, and with these values fixed and for various choices of/3 and c we have carried out least-squares fits to measured total and differential cross sections from 4.5 to 22 GeV/c (ref. [6] ), varying the remaining parameters subject to the constraints

(a)

7B>0;

(b)

Im A ' = 0 near t = - 0 . 2 (GeV/c) 2.

Both (a) and (b) are indicated by FESR's [9,10], and (a) is consistent with the elastic nN polarisation data [6,9] ; (a) is needed because do/dt fixes only the size of B, not its sign, and (b) is equivalent to demanding the crossover effect [11] seen in the elastic angular distributions. Despite the comparative crudeness of the model, a variety of reasonable fits are possible (X2/(deg. freedom) < 4) depending on the chosen values of/3 and c, and on how constraint (b) is enforced. The values of 7B and aB are well-determined by do/dt between t = 0 and the dip at t = - 0 . 5 5 (GeV/c) 2, while the forward and wider-angle data plus constraint (b) are mainly responsible for fixing 7,4, aA and h A . From the fits, predictions of polarisation 9~can be read off for comparison with experiment. With 15= - 1 and c ~ 5 (which are values corresponding to the conventional AE

J. I4;.C o l e m a n a n d R . C.Johnson, R e g g e cuts

617

prescriptions [2,3] ), the equality ct = 0 holds in the fitted energy range for - t = 0.2 to 0.3 (GeV/c) 2, where therefore 9 changes sign. From constraint (a), ~ is positive at small Jtl (in agreement with the older data [ 12]), and so the prediction, typical of the usual Argonne-type models [2], is that 5~ is negative for - 0 . 3 > t > - 0 . 5 5 (in (GeV/c2), disagreeing strongly with the new data [ 1], which show large positive values in this region. If/~ and c are allowed to vary in the fitting, (subject to t3 < - 1 ) , values/~ ~ - 1 . 0 5 and c ~ 0 are preferred. These numbers are determined mainly as a balance between the requirements that the cross-over zero should not move towards t = 0 too rapidly as energy increases, and that do/dt beyond the dip (where the cut is relatively important) should shrink as observed. The first point needs slow pole-cut relative energy variation, and the second requires the cut to go down fairly fast. In this case the equality a = 0 occurs near t = - 0 . 5 5 (GeV/c) 2, giving an approximate double zero, and the sign of 5aagrees with the data. However, as fig. 1 shows, the predicted size of 9 i s rather small.

4-O-5 ~

co =:

4

+7

~

-o5~

,

,

I

_ ~

0

J

'

-0'5

t

,t 8C-eV/c

-I'0 (GeV~e)=

-1'5

Fig. 1. P o l a r i s a t i o n p r e d i c t i o n s o f m o d e l s A a n d M c o m p a r e d w i t h d a t a f r o m ref. [ 1 ] . P a r a m e t e r values are: M o d e l A:

M o d e l M:

3~A = 2 3 . 0 G e V -1 ,

h A = - 7 . 8 G e V -1 ,

3'B = 2 6 4 . 0 G e V -2 ,

a B = 0.7 ( G e V / c ) -2 ;

c~' = 1.05 G e V -2 ,

3'A = 14.5 G e V - l ,

aA = 7 . 3 8 ( G e V / c ) -2 ,

3'B = 2 5 4 . 4 G e V -2 ,

h B = - 3 1 2 . 0 G e V -2 ,

aB = 1.63 ( G e V / c ) -2 .

aA = 6 . 2 4 ( G e V / c )-2 ,

h A = - 1 5 . 6 G e V -1 ,

A n isospin f a c t o r - x / 2 is n e e d e d to give a m p l i t u d e s for n - p ~ ~r0n.

618

J.W.Coleman and R. C.Johnson, Regge cuts

The situation is not much improved by allowing variation of the-parameters b, a c etc., and even if kB is allowed to go free and the polarisation data included in the fitting, the size of (~increases by only a few percent. Thus it appears that small values of ~ are typical of this kind of model with nonsense zeros, although of course it is not impossible that agreement with experiment can be obtained by including more complicated effects than the simple ones considered here. We note that at much larger energies, the prediction is that the two zeros of separate, one (at c~ = 0) remaining fixed in t, the other (at a = 0) moving logarithmically towards t = 0, so giving 9(t) similar in structure to the Argonne model predictions at current energies [2]. (ii) Model M. Another possibility that has attracted some attention [3] is that the pole amplitudes are free of nonsense zeros (N=I in eq. (1)), and the observed structure in angular distribution is geometrical, i.e. due entirely to pole-cut interference. Specifically, the picture of n-p ~ nOn is as follows: (a) the dip-bump behaviour of do/dt results from destructive pole-cut interference in B; (b) cancellation between the pole and a relatively stronger cut term is wholly responsible for the crossover zero in Im A'; (c) the main contribution to ~ is still Im (R~CA) but there is also a sizeable term from Im (C*BRA), and both contain the factor sinln(a-O). Because the cut terms are strong, there is the possibility of a large phase-difference between A ' and B, and thus large values of 9. Fits to the cross section data by varying 7Aa~, aAd~, a' and kA,B subject to the constraints on the sign of B and on the presence of the crossover zero again lead to polarisation predictions that depend on/3 and c. However, if these parameters also are allowed to vary, the preferred values are/3 ~ - 1 . 2 and c ~ 0, which again balance the requirements of reasonable shrinkage plus amplitude zero-structure at roughly fixed t. The resulting predictions of 9 a r e included in fig. 1, and they are in good agreement with experiment both in sign and magnitude. Note that the sign-change in (P, which comes from the equality a = 0, is not at a fixed t-value but moves logarithmically with energy towards t = 0, in agreement with the possible trend in the data. The A ' and B amplitudes at 8 GeV/c are plotted in fig. 2, where they may be compared to the amplitudes for model A. An important point to notice is the close similarity between Re B in each case. In model A it has a double zero at t = - 0 . 5 5 (GeV/c) 2, and in model M there is a minimum (value of 0.2 GeV -2) at t = - 0 . 5 9 (GeV/c) 2. Consequently, existing elastic 7r±p polarisation data (which show mirror symmetry and a double zero) do not necessarily make a clean distinction between the rival dip mechanisms, as is sometimes implied, (e.g. refs. [7,8] ). If/3 is held near - 1 and c increased, the zero in 9 m o v e s inward, the positive maximum decreases, and the negative maximum increases markedly, to give a pre-

J. W.Coleman and R. C.Johnson, Regge cuts

3O

3O

A'

619

B

2O

(a)

o o

-05

-Io

o

-O5

3O

B

2C

2C

(b)

-tO

'°L i

0

-0.5

-tO

0

-0.5

-LO

Fig. 2. AmplitudesA' and B (in GeV-1 and GeV-2 respectively) at 8 GeV/c plotted against t (in (GeV/c)2). Full lines are real parts, and dashed lines imaginary parts. (a) Model A; (b) Model M.

diction similar to that of the usual Michigan model fits [3], (see fig. 12 o f ref. [3] ), which disagrees with the new data [1 ]. We note however that this model predicts the approach of 9(t) to the Michigan structure [3], and the transition from a minimum in Re B to a double zero and then to a pair o f separating simple zeros, logarithmically with increasing energy.

4. CONCLUSIONS The following points emerge: (i) simple models both with and without nonsense zeros can explain the sign of the newly-measured polarisation - however the latter, with its stronger cut contributions, finds it easier to explain the large magnitude; (ii) the crucial feature is the energy-dependence o f the cut (giving its phase through crossing) which has to be such as to maintain the approximately fixed-t structure of the data (dip, crossover), while giving reasonable shrinkage; (iii) although decisively accurate higher-energy experiments will be difficult, model A has the characteristic feature of one zero in 9 f i x e d at ct = 0, while model M predicts a logarithmic movement of its zero towards t = 0 as energy increases. Precise higher-energy elastic 7rN polarisation data would also be useful. We thank Alan Martin for valuable discussion, Fred Gault for giving us details of a Michigan model fit, Gordon Ringland for supplying tabulated polarisation data, and an anonymous referee for his helpful comments.

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J.W.Coleman and R. C.Johnson, Regge cuts

REFERENCES [ 1 ] P. Bonamy et al., contributed paper at the Amsterdam International Conference on Elementary Particles, June 30 - July 6 1971, and to be published. [2] R.C.Arnold and M.L.Blackmon, Phys. Rev. 176 (1968) 2082; G.Cohen-Tannoudji, A.Morel and H.Navelet, Nuovo Cim. 48A (1967) 1075. [3] F.S.Henyey et al., Phys. Rev. 182 (1969) 1579. [4] V.Singh, Phys. Rev. 129 (1963) 1889; W.Rarita et al., Phys. Rev. 165 (1968) 1615. [5] J.B.Bronzan and C.E.Jones, Phys. Rev. 160 (1967) 1494. [6] G.Giacomelli, P.Pini and S.Stagni, Blue report CERN-HERA 69-1 (Nov. 1969). [7] C.Lovelace, Invited talk at the CalTech Phenomenology Conference, March 1971, Rutgers preprint, to be published. [81 H.Harari, Phys. Rev. Lett. 26 (1971) 1400; G.A.Ringland and R.J.N.Phillips, RHEL preprint RPP/C/8 (1971). [9] V.Baxger and R.J.N.Phillips, Phys. Rev. 187 (1969) 2210. [10] R.Dolen, D.Horn and C.Schmid, Phys. Rev. 166 (1968) 1768. [ 11 ] W.Rarita and R.J.N.PhiUips, Phys. Rev. 139 (1965) B 1336; V.Barger and L.Durand, Phys. Rev. Lett. 19 (1967) 1925. [12] P.Bonamy et al., Phys. Lett. 23 (1966) 501.