A global model of K+ photoproduction

Nuclear Physics B66 (1973) 493-504. North-Holland Publishing Company

A GLOBAL MODEL OF K + PHOTOPRODUCTION A.R. PICKERING* Dept. of Physics and Astronomy, University College London Received 24 August 1973

Abstract: We extend the use of resonance-saturated fixed-t dispersion relations to the associated photoproduction process 7P ~ K+A. In contrast with previous analyses we are able to demonstrate that the Born term couplings in this process are compatible with those determined from K+-p data, namely (g~ + 0.8g~)/4~r = 12.2 -+2.5. We find that the distinctive structure of the data at backward angles in the low energy region indicates substantial contributions from the elusive Dl 3(1700) resonance. With the introduction of a very simple model for the imaginary parts of the high-energy amplitudes we are able to fit the 5-16 GeV differential cross section data and obtain a qualitative description of the polarisation data at 5 GeV.

1. Introduction The ability of resonance-saturated fixed-t dispersion relations to provide a consistent description of low energy pion photoproduction has been indicated by the quantitative success of the recent large scale analysis of Moorhouse and Oberlack [1 ]. We now propose an extension of this approach to 7P -~ K+A. This process is of interest for two main reasons: (a) Previous isobar analyses have found rather small Born term couplings [ 2 - 4 ] : G2- 1 2+ 2 < ----4-~(gA 0 ' 8 g x ) - 8 . 0 , in disagreement with analyses of processes involving only strong interactions, which have indicated considerably larger values of G 2. In particular, the most reliable information would seem to be the estimate of Knudsen and Pietarinen from K+p scattering using forward dispersion relations [5], G2 = 1 2 . 2 - + 2 . 5 . (b) It has been suggested that there may be evidence in associated photoproduction for the existence of so-called stray baryonic states [6]. These are states whose * Address after 1st October: Niels Bohr Institute, Blegdamsvej 17, 2100 Copenhagen O, Denmark.

A.R. Pickering, K + photoproduction

494

existence is required by the quark model but which are not seen in ~rN phase-shift analyses. In the application of fixed-t dispersion relations to low energy scattering, the imaginary parts of the amplitude outside the data range are usually approximated by a few pseudo-resonances. We prefer to adopt the alternative approach of approxirnation in terms of a simple Regge model. We are thus in a position to calculate the real parts of the scattering amplitudes at all energies without any risk of Regge-resonance double-counting. Similar models have been used in the study of high energy 7rN and KN chargeechange [7,8] and charged pion photoproduction [9] using the fairly well determined low energy amplitudes for these processes as fixed input. The novel feature of our analysis is that we attempt to fit both high and low energy data simultaneously.

2. Parametrisation We consider the process T(k) + P(Pl) ~ K+(q) + A(P2), and write the usual dispersion relations between the real and imaginary parts of the amplitudes,

ReAl(S, t) = Poles + P

y

ds'

( m +/,t) 2

ImAi(s',t) If s'+ $

(A+tt) 2

du'

ImAi(t'u') , H--U

The structure of the Born terms and invariant amplitudes has been discussed many times before [2,10] and for brevity we do not discuss them here*. We split the dispersion integrals into two parts:

(m+#) 2

. (m+u) 2

+

ds'. ~-

Below g we parametrise the imaginary parts of the amplitudes in terms of resonances, above ~- we use our Regge model. We introduce the resonances via the usual multipole expansion of the invariant amplitudes [10,11 ], and parametrise the contribution of individual resonances to the multipole amplitudes as [ 12],

* Our amplitudes are defined in terms of the gauge- and Lorentz-invariant matrices first defined in ref. [10l.

495

A.R. lh'ckering, K ÷ photoproduction

Im (El+) = _ , - x / q k j v ( ] 7 +1) (MR2-S) 2 + P2M 2

1

J,y = l + 1 ,

rM M t_+

Im (M/±) - ~/clk].~(]~, + 1) (M~ - s) 2 + p2M~

Jr = l .

The notation is entirely conventional. We parametrise the strong decay widths in terms of barrier factors appropriate to the spin of the resonance, F l t = 71±

where 7l± is a reduced width independent of energy. Since such barrier factors have no physical significance away from the KA threshold [ 13] we cut them off at higher energies in a continuous fashion. We simulate the effect of the zrN threshold in a similar manner,

We treat the electromagnetic widths PE,M as constants since the 7N threshold is considerably below the lower limit of the dispersion integral. Although the above remarks specifically concern s-channel resonances, with obvious changes they also directly relate to the u-channel. We now turn to the construction of high energy amplitudes. We first remark that the s-dependence of the differential cross sections for K+A photoproduction above 5 GeV corresponds to an effective trajectory a(t) ~ 0, which precludes any description in terms of simple Regge poles alone [14]. However various authors have shown that it is possible to fit this data in terms of pole + absorptive cut models [ 15-18]. A well-known feature of such models is that the imaginary parts of the s-channels helicity amplitudes show a Bessel-function-like structure in t, and this observation has been extensively discussed by Harari [19], who conjectures that the amplitudes have the structure

where Ak is the absolute magnitude of the s-channel helicity change. Ja~, is the ordinary Bessel function of order AX and ~ h is some function having the same qualitative features, b is a typical interaction radius, around 1 fm. From past analyses we known that K* and K** t-channel exchanges are the dominant contributions at high energy, and from the duality argument of CapeIla and Tran Thanh Van [14] we expect these to be exchange-degenerate, leading to a combined amplitude which is purely real in the u-channel and has a rotating phase in the s-channel.

496

A.R. Pickering, K+ photoproduction

For computational convenience we parametrise a set of amplitudes H n, labelled by the total s-channel helicity flip, which are related to the conventional helicity amplitudesH w as [20]. W

(H1,Ho, H _ I , H 2 )

W

W

, x/~(H1,H2,H3,H4 S--~

W

) .

ao

We parametrise the imaginary parts of these amplitudes in terms of only five parameters, H 0 = 70Jo(bx/~)eatsa(t)

,

H+ = H 1 + H 1 = ~/+Jl(bX/C---t)eats a(t),

H 2 = 72J2(bx/rZ-7)eatsa(t) H

=H 1-H

1 = 0.

Note that for natural parity exchanges even in the presence of absorptive cuts we have H_ = 0 [18]. The residue functions are taken to be constant. The parameter a determines the deviation of the amplitudes from the simple Bessel function form. We take the trajectory to be a straight line passing through the K* and K** masses, a(t) = 0.35 + 0.82t. This completes the description of our parametrisation and we now go on to discuss the fits to the data.

3. Fits We set out to determine what range of Born term couplings is compatible with with the data. We fix the magnetic moments of the proton and lambda at their experimental values, and the A - X 0 transition moment at the value -~x/~#n predicted [3] by SU(3). We allow the strong couplings to vary, using SU(3) only as a guide to their relative signs and magnitude, g A ' ( 0,

g:~ > 0 ,

IgA[ > g ~ .

We fix the resonance masses and widths at the values given by the Particle Data Group [21 ] but allow their couplings G E = (7l_. FE)½ and G M = (7l_+FM)½ to vary. We also allow the five Regge parameters to be determined by the data. We limit ourselves to fitting data in the range [ t [ < 1.0 GeV 2. The motivation for this restriction is as follows. Above 5 GeV the differential cross sections show a characteristic fall-off, roughly as e -3t at large t [22]. We cannot hope to reproduce this in terms of our simple model, and indeed one doubts the validity of any simple Regge model at large t. We therefore make the standard if arbitrary choice of t = - 1 . 0 (GeV/c) 2 as the cut-off. This involves the loss of thirteen differential cross section measurements in the low energy region. Our data set thus comprises 136 differential cross section and 23 polarization measurements in energy range from

A.R. Pickering, K+ photoproduction

497

Table 1 Solution

gA

gG

G2

2 XLE

2 Xtot

1

2 3 4 5

-13.0

+2.6

-12.0 -11.0 -10.0 - 9.0

+3.1 +3.3 +3.7 +4.8

14.0 12.2 10.5 9.1 8.3

6.9 4.5 4.9 5.2 6.7

7.6 5.6 5.1 5.8 11.2

threshold to 1800 MeV, 48 differential cross section measurements in the range 5 - 1 6 GeV, and 7 polarization measurements at 5 GeV*. In our fits we minimise the usual quantity

1

1

where Pi represents either differential cross section or polarization, and Np is the total number of points included the fit. In table 1 we give the Born term couplings and X2t for various fits having values o f g A from - 9 . 0 to - 1 3 . 0 . We also quote the quantity NE

___1 2 NE i=1 where N E is the number of low energy data points. In table 2 we give the resonance and Regge couplings of fit 2. For comparison we give the quark mode 1 predictions for these couplings [1,24] and also estimates derived from the FN, _~N'r widths determined in the Moorhouse-Oberlack pion photoproduction analysis. In estimating G E and G M we have taken the PN* --, KA widths from SU(3) fits to pseudoscalar meson-baryon scattering [25,26]. Note that we have excluded the D15(1670) from our fits in view of clear evidence that it is only weakly coupled to the 7N and KA channels [27]. In figs. 1 - 6 we plot typical angular distributions and excitation curves, taken from fit 2. As suggested by these curves, the values of ×2.E" ~ 4.5 are deceptively high. In general the fits to the differential cross sections and polarizations are completely satisfactory with perhaps two regions contributing disproportionately to X2L.E.. These regions are, (a) ThreshoM-lOOOMe V. The 12 differential cross section points in this region contribute around 1.0 to X2.E.. It is difficult to discern any systematic deviation * The low-energy differential cross section and polarization data are from ref. [23], the highenergy differential cross sections from ref. [221, and the 5 GeV polarizations from ref. [32].

498

A.R. Picketing, K+ photoproduction

~.0

Ey = 1.2 GeV 3.0

2.0

1.0

I

0.0

1.0

0.5

I

I

0.0 cos 0cm

-0.5

Fig. ]. Angular distribution at 1200 MeV from solution 2. -06

-0.5

-O.t, -0.3 -6 a. -0.2 -0.1 / 0.0

I

0,9

1.0

t1.1

I

1.2

I

1.3

EV(6eV) Fig. 2. Lambda polarization at 90° from solution 2. between our fits and the data. In this region the differential cross sections are rising very fast from threshold and it is possible that small errors in the quoted experimental energies could give rise to considerable contributions to X2.E.. (b) 1 3 5 ° E x c i t a t i o n curve. Although in general reproducing the experimental

A.R. Pickering, K + photoproduction

499

1.5

1.0

o b~ Ocrn= 90 °

0.5

0.011 0.9

I 1,1

1 1.3

I 1.5

E.y (GeV)

Fig. 3. Excitation curve at 90° from solution 2. structure of the data at angles beyond 90 ° our predictions are somewhat below the data at 135 °. This failure contributes around 1.0 to X2.E.. We discuss the possible reasons for this in the following section.

4. Conclusions 4.1. Born terms

From table 1 it is apparent that one cannot determine the Born terms in any way uniquely from the existing associated photproduction data, although couplings in the range - 1 2 . 0 < gA < --10.0 and +3.1 < g z <~ +3.7 are favoured. The insensitivity of the fits to the Born terms is due to the large cancellations which can occur in the real parts of the amplitudes between the Born terms and the u-channel resonances. Variations o f g A are compensated on minimisation by changes in the couplings of these resonances, and therefore a more precise determination o f g A requires more detailed knowledge o f the Fy,__,~A partial widths. We note that fits I - 3 correspond to values of G 2 lying within the range determined from K±p scattering.

500

A.R. Pickering, K+ photoproduction

Table 2 (a) Regge couplings 3'o = +0.020 GeV "r+ =-0.096 GeV

"Yz = -0.021 GeV a = -2.35 (GeV/c) -2

b = 4.30 (GeV/c) -1

(b) Resonance couplings (GeV' 10-3) Resonance

Solution 2

Experimental predictions [ 1]

OE

OE

P11 (1470) S11(1500) D13(1520) St1(1670) F15(1690) D13(1710) P11 (1720) P13(1860) F17(2000) D~3(2030)

-0.35 +0.15 +0.59

2"(1385) A*(1405) A*(1520) A*(1670) Z*(1670) I~*(1750) I~*(1765)

+0.02 +7.31 +0.33 +3.91 +5.06 +5.75 -0.17

aM -1.33

+0.33 +1.04 +0.34 +0.02 -0.60

+0.81 +0.07 -0.21 -0.27 +0.44 +0.28 -0.35 -2.92 +0.91

Quark model predictions [ 1,24] GM

GE

-0.60 + 0.3 +1.8 ± 0.9 +1.3±0.2

+0.7 ±0.2

+0.1 + 0.01

+0.1 +-0.01

+5.00 +0.70 0.00 +0.06 0.00

+0.1 -+0.2 -0.63 0.00

0.0

-1,2 -+ 0.1

aM +0.30

0.00 +3.00 +0.70 > 0

+0.40 +0.04 0.00 -0.22 +0.33 0.00

-0.90 +0.50

-8.43 -0.34

4.2. Resonance couplings As one can see from table 2 the resonance couplings in our fits show a considerable level of agreement with the predictions of the quark model. Although the data is not of sufficient precision for us to be able to assign meaningful errors to our determination of these couplings the signs of larger couplings are fairly certain, and we note that, with the exception of the P11(1470) these are all in agreement with the quark model. Since we are not fitting any data in the crossed channel the higher u-channel couplings cannot be considered as more than effective couplings. However the three lowest mass resonances play a distinctive role in the cancellation of the Born terms, and we may regard their couplings as reasonably well determined. It is encouraging that these couplings prove to be in agreement with the predictions of the quark model. In particular the prediction that the excitation of the ~*(1385), which belongs to the 3+ decuplet, is purely MI+ is very definitely confirmed.

A.R. Pickering, K + photoproduction

s1

501 ~

f

J

J

J

f

f

J

f

J

i

10

7

0.5

0.0 0.90

Ocrn= 120 °

I

1.00

I

1.10

I

1.20

I

1.30

E~ ( GeVI

Fig. 4. E x c i t a t i o n curve at 120 ° f r o m s o l u t i o n 2. D a s h e d curve s h o w s p r e d i c t i o n o f solution 2 e x c l u d i n g D13 ( 1 7 0 0 ) .

We consider the very pronounced dip centred around 1160 MeV seen in the excitation curves at backward angles to be clear evidence for the importance of the D 13(1700) resonance. The existence of this state is required by the L-excitation quark model. All of the nucleon resonances belonging to the {70), L = 1- multiplet of this model have been positively identified in nN phase-shift analyses with the exception of the [8,4] D13 state, which the quark model predicts to have a very small elasticity (< 0.05) [28]. The recent isobar model analysis of 7rN - nzrN of Herndon et al. [29] has provided the first concrete evidence for the existence of a highly inelastic D13(1700) state. Further evidence comes from phase-shift analysis of lr-p ~ K0A [27] and rrN ~ KY. [30]. A recent multipole analysis of associated photoproduction by Deans et al. [31 ] strongly suggested that the structure in the backward direction in this process is due to a D13 resonance, obtaining their best fits with a D13(1690 ), F = 100 MeV. We confirm this suggestion in the context of our model, although a somewhat larger width, F ~ 1 5 0 - 2 0 0 MeV seems indicated. In fig. 4 we show the excitation curve at 120 °, comparing fit 2 with and without the D13(1700 ). From thisit is clear that the structure is a very local effect, which it is impossible to reproduce in terms of only the well-established resonances in this region. As mentioned in the previous section the differential cross sections in our fits lie somewhat below the data at 135 °, although reproducing its structure perfectly.

502

A.R. Pickering, K+ photoproduction

We feel that this does not cast any doubt on our conclusions concerning the D 13(1700), since as one can see from fig. 4 the scale of the structure is largely determined by the background to the D13 rather than by the D13 itself. In this region the Born term and u-channel contributions are very large, and the combined amplitudes are extremely sensitive to cancellations between them. In view of this we do not regard these normalisation problems at 135 ° as a serious shortcoming of our model. Following the suggestion of Donnachie [6] we have also looked for evidence of an F15(1700) resonance (not the F15(1690)) but have found no significant evidence for its existence.

4.3. Regge couplings As seen from figs. 5 and 6 we achieve quantitative fits to the high energy differential cross section with the exception of the extreme t-values. We have checked that the contribution of the Regge real parts is small compared to the resonances in the low energy region, and therefore produces no significant local structure, in accordance with physical intuition. We have also checked that the Regge imaginary parts extrapolated to low energies provide a reasonable average of the resonance imaginary parts, and that our fits are insensitive to the energy ~- at which we change from one parametrisation to the other. We cannot, however, obtain quantitative fits to the polarization data at 5 GeV, Our model predicts large negative polarizations at t ~ - 0 . 2 (GeV/c) 2 in agreement with the data, but our predicted polarizations fall off faster than the data as t becomes increasingly negative. This is a typical shortcoming of Regge absorption models [I 5 - 1 8 ] , and the cause is particularly easy to identify in our model. The fits to the differential cross sections are dominated by the by coupling to the unithelicity-flip amplitude, H+. Because of the assumed Bessel function form of this amplitude it vanishes around t ~ - 0 . 7 (GeV/c) 2, and hence the predicted polarization shows a dip or even a sign change in this region.

4.4. Summary We briefly summarise our conclusions as follows: , (i) Large Born term couplings are compatible with the data. More precise determination of them requires more accurate knowledge of the u-channel resonance couplings. (ii) Distinctive structure in the backward direction at low energies is very strongevidence for the existence of the D 13(1700) state required by the quark model. (iii) In the framework of fixed-t dispersion relations it is possible to fit the high energy differential cross section in terms of a simple Regge model for the imaginary parts of the scattering amplitudes.

A.R. Pickering, K + photoproduction

503

3.0

E.y = 8 GeV

n

2.0

'o bC~

1.0

0.0 ~

0.0

I

I

I

I

0.2

0 ./.

0.6

08

1.0

(GeV/c)

Fig. 5. Angular distribution at 8 GeV from solution 2. 1.5

E-y = 16 GeV 1,0 E

¢

b~ 0.5~

L_ 0.0 0.0

I

I

I

i

0.2

0.I.

0.6

0.B

(GeVlc)

Fig. 6. Angular distribution at 16 GeV from solution 2.

j

1.0

A.R. Picketing, K+ photoproduction

504

I a m very g r a t e f u l to Dr. B.R. M a r t i n a n d Dr. R.C.E. D e v e n i s h for c o n t i n u a l advice a n d s u p p o r t in this w o r k .

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] 11] 12] 13] 14] 15] 16] 17] 18] 19] 20] 21] [22] [23]

[24] [25] [26] [27] [28] [29] [30] [31] [32]

R.G. Moorhouse and H. Oberlack, Phys. Letters B43 (1973) 44. F.M. Renard and Y. Renard, Nucl. Phys. B25 (1971) 490. Y. Renard, Nucl. Phys. B40 (1972) 499. T. Fourneron, Ph.D. thesis (Paris, 1972). C.P. Knudsen and E. Pietarinen, Nucl. Phys. B57 (1973) 637. A. Donnachie, Nuovo Cimento Letters 3 (1972) 217. G.I. Ghandour and R.G. Moorhouse, Phys. Rev. D6 (1972) 856. M.F. Coirier, J.L. Guillaume, Y. Leroyer and Ph. Salin, Nucl. Phys. B44 (1973) 157. M. Hontebeyrie, J. Procureur and Ph. Satin, Nucl. Phys. B55 (1973) 83. S. Hatsukade and H.J. Schnitzer, Phys. Rev. 128 (1962) 468. G.F. Chew, M.L. Goldberger, F.E. Low and Y. Nambu, Phys. Rev. 106 (1961) 201. H. Thorn, Phys. Rev. 151 (1966) 1322. H. Harari and Y. Zarmi, Phys. Rev. 187 (1969) 2230. A. Capella and J. Tran Thanh Van, Nuovo Cimento Letters 4 (1970) 1199. C. Michael and R. Odorico, Phys. Letters 34B (1971) 422. J.L. Alonso, D. Schiff and J. Tran Thanh Van, Nuovo Cimento Letters 5 (1972) 27. G.R. Goldstein, J.F. Owens and J. Rutherfoord, Nucl. Phys. B53 (1973) 197. N. Levy, W. Majoretto and B.J. Read, Nucl. Phys. B55 (1973) 493. H. Harari, Ann. of Phys. 63 (1971) 432, R.L. Walker, Phys. Rev. 182 (1969) 1729. Review of Particle Properties, Rev. Mod. Phys. 45 (1973). A.M. Boyarski et al., Phys. Rev. Letters 22 (1969) 1131. R.L. Anderson et al., Phys. Rev. Letters 9 (1962) 131; H. Thorn et al., Phys. Rev. Letters 11 (1963) 433; C.W. Peck et al., Phys. Rev. 135 (1964) B830; B. Borgia et al., Nuovo Cimento 32 (1964) 218; M. Grilti et al., Nuovo Cimento 38 (1965) 1467; R.L. Anderson et al., International Symposium on electron and photon interactions, Hamburg, 1965; S. Mori et al., Cornell thesis (1966); D.E. Groom et al., Phys. Rev. 159 (1967) 1213; A. Bleckmann et al., Z. Phys. 239 (1970) 1; T. Fujii et al., Phys. Rev. D2 (1970) 439; D. Decamp et al., Orsay preprint LAL 1236 (1970); H. Going et al., Nucl. Phys. B26 (t971) 121; P. Feller et al., Nucl. Phys. B39 (1972) 1413. L.A. Copley, G. Karl and E. Obryk, Nucl. Phys. B13 (1969) 303. D,E. Plane et al., Nucl. Phys. B22 (1970) 93. J. Meyer and D.E. Plane, Nucl. Phys. B25 (1971) 428. C. Lovelace and F. Wagner, Nucl. Phys. B25 (1971) 411. R.G. Moorehouse, Glasgow preprint G,U.11 (1973). D.J. Herndon et al., LBL-1065 Rev. (1972), submitted to Phys. Rev. W. Langbein and F. Wagner, Max-Planck preprint (1972). S.R. Deans, D.T. Jacobs, P.W. Lyons and D.L. Montgomery, Phys. Rev. Letters 28 (1972) 1739. G. Vogel et al., DESY preprint 72/26 (1972).