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Polarization predictions for K+ photoproduction
Nuclear Physics B53 (1973) 197-216. North-Holland Pubhshing Company
POLARIZATION PREDICTIONS FOR K + PHOTOPRODUCTION*
G.R. GOLDSTEIN, J.F. OWENS Ill and J. RUTHERFOORD Department of Physws, Tufts Umverstty, Medford, Mass. 02155
Recewed 5 June 1972 (Revised 19 October 1972)
Abstract: Using the existing data for charged plon photoproduction, some general features of the observables m K+ photoproduction are obtained. Using the constraints imposed by duality allows further predictions to be made. Two versions of the Reggelzed absorption model are presented and comparison is made with earher models.
1. Introduction The data currently available for the reactions 7p~K+A,
(1)
3' P ~ K + ~2°
(2)
at high energy are hmited to measurements of the differential cross section [ 1]. Experiments which measure either the polarized photon asymmetry or the recoil lambda polarizahon are now in progress. It is therefore of interest to consider the general features that may be expected for the observables that are to be measured. In this analysis we shall assume the existence of Regge poles and Regge cuts, although the precise form for the latter is not required. As shown in table 1 there are six Regge trajectories which may contribute to reactions (1) and (2). Even when the number of free parameters in a particular model is very limited the fit to the existing data need not necessarily be unique and the predictions for the various observables are certainly model dependent. It is nevertheless possible to gain some insight into the structure of the various K + photoproduction observables by considering the existing data for charged pion photoproduction. * Work partially supported by Atomic Energy Commission.
198
G.R. Goldstem et al., K + photoproductton
Table 1 Exchan{es m charged-plon and -kaon photoproductlon. KA (KB) denotes the strange member ofthe.P (~ = 1++ (1 +-) nonet. A ( 2 - ) and K ( 2 - - ) denote the Y = 0, I = 1 and Y = 1,I =1 members of the (unob~rved) J/'C = 2 - - nonet Amplitude
3' P ~ *r+n
3' P ~ K+A
f+
A2, O
K**, K*
f+
At,A(2 ' - )
KA, K ( 2 - - )
.t'~"
A2, a
K**, K*
f~
n,B
K,K B
Using SU(3) and tile current estimates for the appropriate d / f ratlos at the baryon vertex one may obtain information on the relative strengths of the various Regge exchanges. This results in several predictions for the general features of the K + photoproduction observables. By also including the constraints of exchange degeneracy and vector dominance we can make some additional, though weaker, statements about these observables. In sect. 2 we give expressions for the various observables and discuss the tmpllcatlons of the charged pion photoproductlon data. Amplitude relations required by duality are given and some possible effects of these relations are discussed. In sect. 3 we contrast the predictions of two forms of the Reggelzed absorption model. In sect. 4 we summarize our predictions and present some concluding comments on the various models for K + photoproduction.
2. Observables In order to discuss the observables for reactions (I) and (2) it is convenient to define s-channel helicity amplitudes with definite parity in the t-channel to leading order in s. We define the following set of s-channel helicity amplitudes f l = f l + , 0+' = J 1+,0- ,
(3)
f3 f4 = f l - , 0 - ' where fabcd is the amplitude for a photon of heliclty a striking a target proton with
199
G.R. Goldstein et aL, K ÷ photoproducnon
helicity b and producing a spin-zero meson and a nucleon with helicity d. Then we may define the following combinations +
f i = ~- ( f l -+f4) ' r2 =
(4)
of2
where the superscript + ( - ) indicates natural (unnatural) parity exchange in the tchannel. In calculating the observables we use the density-matrix formalism [2, 3]. The density matrix for a plane-polarized photon whose electric vector makes an angle ¢'r with respect to the scattering plane is
p'X= ½ [
]
-e-2i'3'
_e2ig"r
(5)
1
Now consider the helicity frame of the recoil baryon, i.e., the rest frame of the recoil baryon in which the z-axis is in the direction of the outgoing kaon and the y-axis is in the direction kv X k K (the normal to the scattering plane). In this frame the (unnormalized) recoil-baryon density matrix is given by
'ram'
(*,)-- ½ 4 ,-m 'oo"
m'"
(6)
The recoil baryon polarization in the ith direction (i = x, y, z) is given by
A, (¢v) = (p(~ov) o i) / (p(~pv)),
(7)
where o~ is the ith Pauli matrix and < >indicates a trace. Using eqs. (4) - (7) we obtain the expressions +*
A x(•.r)=-sin2*vlm
_
Lf1 f2 - f l
*
f221 /(p(gv) >'
(8a)
Ay (~p~,)= Im [f~l* f~2 + f l * f 2 - cos 2 , ~ (f~l*.f~2 - f l * f ~ ) i / < a ( , v ) ) ,
(8b)
A z (gv) = - sin 2 9v lm [ f l * f~l + f2- f2"* l/(P(9.r)),
(8c)
=½[1,t~] 12+l.~212+~[12+[f212cosR~p,y(~]
12+~212-[f112
(8d)
-br2 12)]. The amplitudes are normalized such that do
(~P.r) = ~
~
) <#(~P,),) •
(9)
200
G.R. Goldstem et al., K + photoproductton
The polarized photon asymmetry is gwen by
Y, = [do/dt (90 °) - do~dr ( 0 ° ) ] / [ d o / d t (90 °) + do/dt (0°)] :([q[2+[f~I2--~fl12--[f212)/([f([2+[.f;[2+[fl
[2+[f212 ).
(10)
In order to obtain predictions for the structure of the K + photoproduction observables from the charged-pion photoproduction data we must first make some assumptions concerning the helicity amplitude structure. We shall assume the existence of the evasive Regge pole exchanges hsted in table 1. Furthermore, we shall assume that there is a Regge cut associated with each Regge pole and that its structure is consistent with that given by the absorption model with non-flip elastic-scattering corrections. Thus, a cut which contributes to.f 1 will give an equal contribution to f4 up to an overall sign which is determined by the parity of the associated exchanged particle since both f l and f4 have overall hel,city flip equal to one. This means that cuts m f l and J4 contribute to either t~1 off1- but not to both, i.e. they have definite parity. However, f2 and f3 have overall helicity flip zero and two respectively so that their cuts are not simply related. Thus, a cut in.f2 a n d f 3 contributes to both f~" and f ~ and hence does not have definite parity. Moreover, the cuts in ]'2 do not vanish at t = tram s,nce absorptively generated cuts satisfy the t-channel conspiracy relation*. However, that portion of the cut which does not vanish in the forward direction has an extra factor of (In v + constant) in the denominator and will hence forth be called the "non-leading" cut contribution. Note that with this form for the helicity amplitudes only.f2 Is non-zero in the forward direction and h e n c e f ~ a n d f ~ become equal at t = tmm. The residue for the exchange of a strange meson in K+A photoproduction is related to the residue for the appropriate non-strange meson m lr+ photoproduction v/a SU (3) by /3 (A) = - / 3 (70 (3 f + d ) / x / ~ ( f + d).
(11)
Therefore, if all the d / f ratios are greater than - 1 or less than - 3 there will be no relative sign changes among the residues in going from n + to K ÷ photoproduction. Furthermore, if exchange degeneracy and duality are approxtmately correct, exchange-degenerate pairs of trajectories have approximately the same d / f ratios in their residues. Hence the ratio of the residue of an even-signature trajectory to the residue of its odd-signature partner is approxtmately the same for the exchange-degenerate pairs of non-strange exchanges in n + photoproduction as for the corresponding pairs of strange exchanges in K÷A photoproducnon (see table 1, where, for example, the p and A 2 residues in zr+ will have approximately the same ratm as the K* and K** residues in K+). This latter statement is not true for either rrphotoproductlon, where the ratm of an exchange-degenerate pair has a relative minus sign compared to 7r+ or K ÷, or for n °, where only one signature of the pair * Appendix B(e) of ref. [5] contains a discussion of the physical interpretation of this conspiracy relation.
G.R. Goldstem et at, K ÷photoproduction
201
can be exchanged (e.g. p not A2, B not n) and lsoscalar exchanges contribute as well. n + photoproduction is thus the more appropriate reaction for comparison. It should be emphasized however that d / f ratios for the vector-meson nonet, the pseudoscalar nonet, and the A t nonet are not required to be equal, so that the relative strengths o f the non-strange members of these multiplets can be qmte different from the strengths o f the strange members (e.g. the ,opn/rrpn ratio of couplings need not be the same as K * p A / K p A ) . Furthermore the large mass splitting between n and K will significantly depress the strength of K exchange m K + A photoproduction compared to n exchange in n + photoproduction (if the d/f ratios are ignored) since the n pole is much nearer the physical region for the latter reaction than the K is in the former reaction. We will state this more precisely in the following discusslon, but now simply stress that n + photoproduction alone does not suffice to predict K + A - more information on couplings and mass sphttings must be used. The ratio o f the residues appearing in K ÷ zo and K + A photoproduction is gwen by /3 ( Z °) //3 (A) = x / 3 ( f - d ) / ( 3 f + d).
(12)
Eq. (12) shows that there will be a sign change for the residues ifd/fis greater than 1 or less than - 3 . With the foregoing definitions we can now discuss the imphcations of the n ÷ photoproduction observables.
2.1. Polarized photon asymmetry From eq. (1 O) it can be seen that Z measures the relative amounts of naturaland unnatural-parity exchanges. Z for 7r+ photoproduction is characterized by a rapid rise to Z = + 1 at t = -mTr2 and then a gradual fall-off to Z "" 0.7 for t "~ - 1.0 GeV 2 [6]. Now, for small t, Z is dominated by f~ since both fll and f i - go to zero as ~/"S}-while f~2 and/~- get contributions from f2 which need not vamsh provided that the conspiracy relation is satisfied. Since Y = + 1 this then ~mpiles that fl~- = 0 at t = - m~r. 2 One method for achieving this is pole-cut interference where the unnatural-parity portion of the n cut cancels the unnatural-parity n pole leaving the natural-parity portion of the cut. Now consider K÷A photoproductlon where the counterpart of n exchange is K exchange. Usingd/f= 1.5 17], eq. (10) shows that the n and K residues should be about equal in magnitude. However, the K has a larger mass than the n so its contribution to the amplitudes should be approximately reduced by m2/m 2 at t = O. In addition, the point at which f~- vanishes should now be near t = -ml~. We thus expect to see )2 large and peaked near t = - m 2 for reaction ( I ) although it may not equal 1 due to the influence o f other terms which are s~zeable near t . . . . m2K. For reaction (2) we expect a somewhat different behavior. Eq. (11) shows that K exchange is suppressed in K ÷ Z ° photoproductlon using the same d/f ratio as above. Thus, the peaking near t = --m2Kwill not be present and Z will rise more smoothly
202
G.R. Goldstem et al.. K+photoproductton
to a large positive value. Th~s latter pomt follows since the natural parity K* and K** exchange are now dominant just as A 2 and # exchange would dominate n + photoproductlon if n exchange were suppressed.
2. 2. Polarized target asymmetry The polarized target asymmetry, Ty. for a target polarized perpendicularly to the scattering plane has been measured for n ÷ photoproductlon using unpolanzed photons [8] and has been found to be negative. The expression for T v is given by eq. (8b) with the opposlte sign for the f i - terms. Assuming for the moment that f i - is small then Tyis a measure of the interference between f~l and f~2. This interference results primarly from the A 2 term in f~l and the natural-panty parts of the n and A 2 cuts inf,2. Thus, the relative sign between the rr and A 2 residues is determined [9]. The d/f ratios for K** exchange have been determined in refs. [10, 11]. For the non-flip nucleon vertex d/f~- - ½ and for the flip vertex d/f~- 3.5. Thus, the K and K** d / f ratios are all greater than - 1 and hence the relative K - K * * residue sign in reaction (1) is the same as the relative 7r-A 2 residue sign in 7r+ photoproductlon. Therefore, Ty should be negative in K÷A photoproduction. The situation is different however for K ÷ y o photoproduction since now the flip K** and K couplings change sign while the non-flip K** does not. Therefore, there should be an overall sign change for Ty in reaction (2). Ty has been measured for reactions (1) and (2) but without distinguishing between the A and ~o [8]. Since the K÷A cross section ~s larger than that for K+~ ° it is reasonable to suppose that this measurement of Ty is dominated by reaction (1). The measured Ty is negatwe which is consistent with our expectations. Furthermore, if f i- is small, as will be discussed below, then Ay ~- Ty so that Ay will also be negative. Having obtained predictions for the general features of ~ and Ty we shall now consider the effects of exchange degeneracy and vector dominance in order to determine whether or not there are any further distinctive features that may be expected for the observables in reactions (1) and (2). Consider the reactions K-p ~ V° A V ° p - , K+A,
(13) V° = p , ~ , ¢,
(14)
which are related by line reversal. As noted in ref. [12] the quark duality diagram * This means that the pole terms for K - p --, p°A and K-p ---*toA will yield a purely-real phase while K - p ~ $A will have a phase e-tna. We have used vector dominance to predict these three differential cross sections from the fit to "yP--, K+A. The resulnng CA cross section is larger than the ,o°A and wA cross sections in disagreement with experiment [ 13 ]. This same disagreement is seen in other pairs of line-reversed reactions such as KN -, nA and ~rN ~ KA where the absorption model predicts the reaction with the rotating phase to be larger than the reacUon with the purely-real phase [ 14].
203
G.R. Goldstein et al., K + photoproduction
for K - p -~ p°A is an improper one and hence the imaginary part of the amplitude for this process must vamsh in the high-energy limit. The same comment * holds for K - p ~ ~oA and for ~ p -~ K÷A. Satisfying these relations puts constraints on the K*-K**, K - K B , and K A - K ( 2 - - ) couphngs in reaction (14). Vector dominance for p, ~ , and ~0 then allows us to combine the results from these three reactions to obtain a set of constraints for the amphtudes m K ÷ photoproduction. The resulting relations are Imf(K*) = ~ Imf(K**),
(15a)
l m f ( K B ) = ~- l m f ( K ) ,
(15b)
lmf(K (2--)) =] Imf(KA).
(15c)
Furthermore those relations imply that the trajectories are pairwise degenerate for K * - K * * , K B - K , and K ( 2 - - ) - K A. The latter pair of trajectories have states with jPC = 0 - - , i +÷, 2 - - , . . . . Of these the first is forbidden in a quark model m which mesons are constructed from q~l pairs. To date no 0 - - particle has been observed [ ! 5] so the exchange-degenerate K A K ( 2 - - ) residue must have a zero at a = 0. This, however, is not the case for the K - K B exchange-degenerate residue since the K pole is present at a = 0. Therefore, we expect the K A - K ( 2 - - ) terms to be suppressed relatwe to the K - K B terms by a factor o f a . The main features of K ÷ photoproduction will then be determined by the K * - K * * and K - K B trajectories. The exchange-degeneracy requirements (15) and the absence of poles in the physical region can be used to mfer the existence of nonsense wrong-signature zeroes (NWSZ) when cut effects are not present. These will occur at a = 0, - 2 , - 4 for the K * ; a = - 1 , - 3 , . . . for the K and K**; and at a = - 2 , - 4 . . . . for the K B. In the following discussion we shall assume the existence of this NWSZ structure ** 2. 3. Differential cross section
Experimentally a dip is observed in the forward direction contrary to chargedpion photoproduction. This is accounted for by the fact that the larger K mass causes the K pole to be less important in K + photoproduction as stated above. In addition, since g~N n > g 2 N x [7] the K will be even less important in K+Y:0 photoproduction. This is consistent with the data which show a more pronounced dip m the latter case. As - t increases K** exchange will become dominant. Since there is no NWSZ for K** exchange for - t < 1 this accounts for the lack of a dip at the NWSZ of the K* [121. * See footnote on facing page. ** An alternate point of view is that the entire (pole + cut) amplitude satlshes the relations (16) so that NWSZ's are not required. See sect. 3 for a more detailed discussion of such a model.
204
G.R Goldstem et al., K + photoprocluction
2•4. Polarized photon asymmetry Recent data for r r - p ~ r/n [16] indicate that there is a dip at t = --1.5 GeV 2 which could be attributed to the A 2 NWSZ. The K*--.K** trajectory lies somewhat lower than the o - A 2 trajectory so that the position of the NWSZ may be moved in shghtly. The exact location depends on the details of the K * * - P cut and how much it displaces the zero m the pole term. The differential cross section data for K ÷ photoproduction is smooth and featureless m the t region - t ~< 2.0 GeV 2 so the K*, K, and K B terms plus the various cuts must fill in the unwanted zero in the K** term. However, the vanishing of a large natural-parity contribution should have an effect on ~, causing a dip at the position of the NWSZ. Thus, measurements of Y~ in the larger t region 1.2 - 1.5 should show a dip. This dip will be more pronounced in reaction (1) which has the large unnaturalparity K-exchange contribution.
2. 5. Recoil baryon polarization From eq. (8b) we see that Ay(O Q) is a measure of the interference between fiand f~-. We expect that the terms lnfi- which are due to Kg, K ( 2 - - ) and their cuts will be small relative to the other amplitudes and hence predict Ay(O ) to be small "l'. For the case of Ay measured with unpolarlzed photons the small t region is dominated by the K** term mffl interfering with the K * * - P and K - P non-leading cut terms i n f f2. The non-zero value for the cross section at t = tmm indicates substantial values for these terms. Hence, we expect large interference effects especially near the forward direction. The phase of the non-leading cut terms is more slowly varying than that for the poles so there will be a point where the phases become nearly equal. Here IAyl will show a minimum. Thus, we expect large negative values for Ay at small t followed by a fall off to smaller negative values at larger t. From eq. (8a) we see that A x is a measure of the interference b e t w e e n f f1 and f ~ . For small t, f~- -~ff2 and hence A x ~ Ay (4~3, = 45 ° is tmphed when no other value of q~ is specified for A x or Az). As - t increases however, f~- deviates from f2 since it contains the K-pole term which has a NWSZ near t = - 1.0 GeV 2. This NWSZ will cause A x to have a zero near t = - 1.0 GeV 2. Eq. (8c) shows thatA z is dominated by the interference between ~ and f~-. Therefore, A z should also have a zero near t = - 1.0 GeV 2 due to the presence of the K NWSZ. Finally, in going from K+A to K+E ° photoproductlon the non-flip K** coupling does not change sign while the fhp K** and K couphngs do, provided that the current estrmates for the d / f ratios are approximately correct• Then f2 and f~- wdl change sign and fll will not. The Ay and A x will change sign while A z wdl not. •
O
J Recent evidence has been reported [ 17] whtch indicates possible K-K A interference in reaction (14). This would imply ]'i- ~ 0 m reactions (1) and (2). However, we expect this effect to be small for the reasons discussed previously
G.R. Goldstem et al., I ~ photoproductton
205
3. Regge cut model predictions In this section we shall consider the predictions from two different models both of which consist of Regge pole exchanges and Regge cuts generated by absorptive corrections. The two models differ in the manner m which the exchange degeneracy relations (15) are employed. In the first model it is assumed that the Regge pole amphtudes satisfy the relations (15). The residues then must have the NWSZ structure discussed in the previous section. When only elastic rescatterlng corrections are employed this model is referred to as the Argonne or "weak-cut" model. If the cut strengths are allowed to become appreciably larger than those Inferred from elastic rescattermg then the term " m t x e d " model is sometimes employed. In the usual apphcations of this model, however, it is the residue structure rather than the cut strength which differentiates it from other approaches. In the second model which we shall conslder it is assumed that if the relations (15) are satisfied then they are satisfied by the entire amplitude (pole + cut). The resldues are therefore not required to have the NWSZ structure. As In the first model the Regge cuts are generated by rescattering corrections but here It is assumed that a number of intermediate states contribute coherently thus yielding a cut strength that may be several times larger than the cut strength from elastic rescattering only. This model is usually referred to as the "strong-cut" model or SCRAM. In applying both of these models to reactions ( i ) and (2) we shall neglect K A and K(2- - ) exchanges. While these exchanges may be present we do expect that their contributions will be small as has been discussed previously. Neglecting these terms will allow us to concentrate on the major features resulting from the larger contributions of the other exchanges. Furthermore, m applying the strong cut model K B exchange will be neglected. This is consistent with the neglect of B exchange m chargedpion photoproduction 1.. The input K* and K** Regge poles for each model have the general form f l = f4 = T~l 17R vaR A ,
;2
=
%
=
_,,2 -
v °R mp+m H
R = K*,
K** ,
(16)
where v -~-l (s--u), A2 = - t + tmi n, Mp = nucleon mass, and Mi! = A or Z ° mass. 7 R and 72R are the residue functions and r/R contains the pole and the appropriate signature factor.
t For a comparison of these models in ~r° photoproductlon see ref. [ 19[. See ref. [20] for a review of Regge-cut models m general.
G R Goldstem et al.. K+ photoproduct:on
206
Table 2 Residue and signature factor structure for the pole terms in the mixed- and strong-cut models The reduced residues 3 K and 3 K are defined in the same manner as 3 K and B~ L Z 1
Residue and signature factor
Mixed model 1-e
K*
Strong-cut model K*
tlraK*
-
2 #l
K** -gl ~K**
sin nCXK,, -
K K 3" n
3'
3K (aK+ 1)
KB KB n
ha,K. *
inc~K
2
2
A +m K
26K e-~in~K
sin ~ro~K
,rd K
"-InaKB
KB 3
-'=:n~K*
K** - ~ln&K, . -2#1 e
l + e -lrraK**
l+e
t
te
ha'K* A2 +inK.2
slnnaK,
K** K** 3'1 r/
'1
(aKB+I)
l - e ~ sin ~raKB
The K and K B pole terms are gwen by
f l =f4 = 0 , 12 --13 =
R = K, K B .
The two models have different forms for the shown m table 2. The n o r m a h z a t l o n factors for chosen so that the reduced residues, 3g , are the duced residues are then assumed to be constant. The first-order cut corrections are calculated a m p h t u d e of the form
(17) residue and signature factors as the strong-cut model have been same for both models. These reusing an effective elastic-scattering
JP = ik W °t°t e --~A=Rp-
(18)
4rr t
t
whereRp= 2 a p ( l n v - ½in)+Cp. W e h a v e s e t a p = O . 4 G e V - 2 andCp
=
4.91GeV -2 by fitting the slope of the n N elastic-scattering cross secuon. We have used Oto t = 25 mb as an estimate for the initial- and final-state scattering cross sections. The resulting cuts have been enhanced by multiphcative " k " factors which take into account the lack of knowledge of the K+A and K+E ° total cross sections as well as possible interme&ate-state contributions. For the m i x e d model there are two such factors, one for the K* and K** cuts and one for the K and K B cuts "~. For the
+ We have "also obtained a fit with one k factor only m the mixed model. The resulting value was h = 2.0 and the cross-secUon fit and predictions were only shghtly changed.
G.R. Goldstein et aL, K + photoproductton
207
Table 3 The parameter values obtained for the mixed- and strong-cut models. The underhned parameters were fixed or constrained as indicated Parameter
Mixed model
Strong-cut model
01K*
0.0678
0.1025
K*
0.2293 K* 3 01
0.4527
02 K** 01 K**
0.1927
02
3 BE*
0.8506
0K
03.
0.3
0KB
0 33 ~K
K* k.~+
1.79
2.61
~*
K* ~_+
1.oo
K** M:÷ K** X+_
~K* ~K*
3.00
hK__
3.94
2 90
(d/.D+K+ *
-0.546
(d/f)K+*_
4.22
(d/]) K_
1.5
2.05
-0.236 5 73
strong-cut model there are five ~, factors, one each for the K* and K** non-flip and flip baryon vertices and one of the K cut. For both models the K* and K** trajectories were taken to be equal and were constrained to pass through the two masses with the result aK,_K** = 0.37 + 0.8 t. The K - K B trajectory was given the same slope and was constrained to pass through the K mass glvlngaK_KB = - 0 . 2 + 0.8 t. The reduced K-exchange residue was calculated by extrapolating to the particle pole and comparing with the Born term. Using the valueg2NN/4n = 15 and Kim's value d / f = 1.5 [7] we obtain the value shown m table 3. As a result of the relations (15) we have taken the d / f ratios for the K* and K** to be the same and those for the K and K B to be the same. Imposing the relations (I 5) on the pole residues then leaves 6 parameters for the mixed model. There are 1 1 parameters for the strong-cut model since these relations cannot be used for the strong-cut model and since there are more ~. factors. The parameters for both models resulting from a fit to the differential cross sections for reactions (1) and (2) are shown in table 3.
G.R. Goldstein et al., K+ photoproductton
208
1.0
I
I
I
i
I
I
I
I
I
0.I
b
-o 0.01
0.0011 0.0
I
),p--
KoA
o
5
GeV/c
A
B
GeV/c
V
II
GeV/c
x
16
G e V/c
I 0.2
I
I 0.4
I tX (GeV/c)
I O. 6
I
I 0 8
I
I I. 0
Fig. 1. The differential cross section fit to "FP ~ K÷A[ 1 ] at 5 GeV/c (0), 8 GeV/c (A), 11 GeV/c (v) and 16 GeV/c (X). Both models give virtually the same fit Table 4 Fitted d/f ratios for the mixed- and strong-cut models and the resulting coNN/pNN c o u p h n g ratios and to and p fltp/non-fhp ratios
d/f
(w/p)N~
++
+
mixed
-0.546
SCRAM
-0.236
Flip/non-flip
++
+ --
p
o9
4.22
7.8
- 0 234
11.5
-0.344
5.73
4.25
-0.405
8.8
-0.845
-
-
209
G.R Goldstein et al., K + p h o t o p r o d u c t i o n
1.0
i
!
'
.
I
,
,
.
.
0.1
f
=L
b "1o
yp ~K*~.o
0.01
0.001
I
O0
.
o
5
G eWc
A
8
Ge V / c
9
II
GeWc
x
16
GeV/c
,
0.2
.
,
0.4
I
(GeV/c)
I
0.6
I
i
0.8
I0
Fig. 2. The differential cross section fit to "tP ~ K+Z ° at the same energies as in fig. 1. The sohd line Is the mixed model and the dashed line is the strong-cut model. Figs. 1 and 2 show the fits by both models to the differential cross sections for reactions (1) and (2). The data are adequately fit in the region A ~ 1.0. However, b o t h models need relatively large cut strengths to fit the data near A = 0 v,qth the result that there is a dip near A = 1.1 in both models due to excessive pole-cut interference. This dip is not present m the data and hence if either o f these models are to w o r k for A > 1.0, higher-order cut corrections will be needed. There is also a small dip at A = 0.25 at the lower energies for K+A. This is due to pole-cut inter-
210
G.R. Goldstein et al., K + p h o t o p r o d u c t i o n
1.0
i
!
i
•
w
w
•
i
(3.8
|
~
0.6
0.4
0.0
I
0.0
0,2
=
i
0.4 0.6 •', (G eV/¢)
i
i
0 8
1.0
Fig. 3. Mixed model predictions for the polarized p h o t o n asymmetry at 5.00 G e V / c for ,3ap~ K + A ( - - ) and 35) --' K + ~ ° ( . . . . ).
1.0
0.8
0.4 ~
0.2 0.0
0.0
0.2
0.4
0.6 t, (GeV/c)
0.8
1.0
Fig. 4. Strong-cut model predictions for the polarized photon asymmetryat 5.00 GeV/c for "rP-* K+A ( - - ) and ",(p--*K+~;° (---).
G.R. Goldstein et al., K+photoproduct~on
211
0.6
/
0.4
\
/
0.2
/ t.i
/
/
/
fie
0.0
hi
'5 <
t
-0.2
i
-0.4
-0.6
-0
8
i
0.0
I
0.2
i
I
I
O.4
I
0.6
i
I
0.8
l
1.0
& (Ge V/c)
Fig. 5 Mixed model predictions for A x ( - - - ) , A y ( - - ) and Az(-.-) at 5.00 GeV/c for 3'P ~ K+A. ference m the K amplitude and is a further indication that modifications to both models are necessary before they can give a truly-detailed description of the data. On the other hand, it is encouraging that the plon-nucleon coupling constant together with Kim's value for the pseudoscalar meson d / f ratio yields a reasonable value for the K residue and for the T./A ratio at small values of A. The fitted values of the vector-tensor d / f ratios are consistent w~th the various estimates given in ref. [ 10]. Table 4 shows the resulting wNN/pNN ratios as well as the p and w flip/non-flip ratios. Figs. 3 and 4 shows the mixed and strong-cut model predictions for the polarized photon asymmetries for reactions (1) and (2). Both models show the qualitative behavior discussed m sect. 2. For both models the asymmmetries show only a slight energy dependence between 5 and 16 GeV/c with the higher-energy asymmetries being slightly more positive due to the fall-off of the K and K B contributions. Figs. 5 and 6 give the mixed model predictions for Ax, Ay and A z for reactions (1) and (2) respectively. Ay shows a large maximum followed by a fall as is discussed m sect. 2. The magnitude of the maximum is reduced for 719 -* K+E ° since the K
212
G R. Goldstem et aL, K + photoproduction
ot
0.8
e) 0 . 6 w I..-
0.4 CO
0.2
0.0
-0.2 0.0
0.2
0.4 A
0.6 (Ge V/c)
Fig. 6. Mixed model predictions for A x ( - - - ) , A y ( - - ) 3'I) -* K+~ °.
O. 8
1.0
and A z ( - . - ) at 5 00 GeV/c for
amplitude is suppressed. Both A x and A z have zeroes near A = 1.0 which result, m part, from the K NWSZ. However, since these zeroes are present m both reachons there must be additional interferences present. All of the curves show only a shght dependence in going from 5 to 16 GeV/c. Figs. 7 and 8 show the strong cut model predictions for Ax, Ay and A z m reactions (1) and (2) respectwely. Here A x is essentially the same as for the mtxed model but Ay now has a zero at A = 0.4. This Is due to the different structure of f l with its stronger cuts and the lack of a NWSZ for the K* A z is large and positive as m the mixed model but there is no longer a zero near A = 1.0. Finally, A x ~- Ay for small A in all cases as expected. The d / f ratios for both models are consistent with those in refs. [ I 0, 11 ] and as a result A x and Ay change sign while Ay does not m going from K+A to K+Z ° photoproductlon. The precedmg discussion shows that both the strong and mixed models give virtually identical fits to the differential cross sections and differ only m some predictions for the polarizations. This is a result of the fact that the dominant exchange, K**, has no NWSZ in the region considered and hence is the same for both models. The different predictions for the polarizations result from the K and K* NWSZ which appear in the mixed model but not in the strong model. There are, however, very few constraints on the large parameter set in the strong model so that
G.R. Goldstein et aL, K + p h o t o p r o d u c t i o n
213
O.B
/
06
/
/
/.
0.4
u~ 0.2 W
I
n,* I-w
0.0
./
"\//w
°/
./11
I "
.~ - 0 2
-0.4
i
-0.6
0.0
0.2
0.4
m
0.6 6. (GeV/c)
Fig. 7. Strong-cut model predictions for A x ( - - - ) ,
Ay(--)
O.B
and Az(-
0
-)
at 5 00 GeV/c f o r
~,p ~ K + A .
these predictions are not unique. As experimental data becomes available the flexibility of this model could result in different fits to the polarization asymmetries while keeping the same differential cross section fits. One further comment can be made concerning the various cut strengths obtained for both models. ;~ = 1.5 corresponds to complete absorption of the lowest pamal waves. Thus each model has cut strengths that correspond to some over-absorption. However, as has been pointed out in refs. [2 l, 22], it Is possible to obtain sundar results without over-absorption if a sharper cut-off is used for the absorbing amplitude. Therefore, over-absorption In these models need not be considered unphysical, but rather can be taken as an indication of the need for a somewhat different absorption profile.
214
G.R. Goldstem et al.. K + photoproduction
o8t 0.6
•
f
-
\
0.4 ¢Y)
I-. tad
0.2
).-
\\N
""
~ \ \ \
~
,
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oo
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\X
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-0.4
-0.6 o o
i
~
2
,
i
o 4
i
~o~
i
Fig. 8. Strong-cut model pre&ctlons for A x ( - - - ) , A y ( - - ) yp --, K+zo.
i
i
08
i
1.0
and A z ( - . - ) at 5.00 GeV/c for
4. Comments and conclusions There are two additional Regge-cut models for K+A and K÷E ° photoproductlon whose predictions may be compared with those presented in this paper. Borgese and Colocci [23] have presented a model consisting of K* exchange and a K*pomeron cut. They obtam good fits to the cross-section data and gwe predictions for the polarized photon asymmetry. Their model predicts a dip in Y, when a K , = 0 smce then only the cut terms are present and hence the natural-parity contribution has been decreased relative to the unnatural-parity part. Capella and Tran Thanh Van have proposed a model [12] with equal amounts of K* and K** exchange as well as the corresponding cut. Because the K* and K** residues are equal the NWSZ's do not lead to dips in the cross section and they obtam a good fit to all the cross section data. Furthermore, since they have chosen a phase for the cut which ~s different from that m the absorption model + the d~p at ~"The cut m ref. [ 12] Is a fixed cut whereas the absorption model gives a moving cut, l.e the phase is a function of t
G.R. Goldstem et al.. K+ photoproductton
215
A > 1 does not occur. Their predictions for E are similar to ours except that a dip at a K ** = - 1, which might be expected if the relations (15) were used, is not seen. The prediction for A y does not decrease as sharply as ours; a consequence of the different cut phase. Furthermore, they predict a zero at A2 = + 1.2. Berger and Fox [24] have also considered various model predictions for the recoil baryon polarization in reaction (2). However, their predictions are for doublecorrelation experiments where e~ther the photon or the target proton is polarized and the recoil-baryon polarization is measured. Thus, their predictions cannot be compared with ours. As indicated in the prewous section, some modifications to the cut will be required in order to remove the dip in the differential cross section for A 2 > 1. Recently, several papers have appeared which propose alterations to the cut phase and/or form in order to account for observed discrepancies in pseudoscalar mesonbaryon scattering [ 14, 25, 26]. In each case the major result is that the cuts are rotated counter-clockwise in the complex plane. The predictions for the general behavior of the A~ gwen here are not sensitive to this phase and hence are unchanged. However, the detailed shape of the A~ wdl of course be quite dependent on such details and will be different for each proposed modification. Furthermore, such cut rotations should decrease the pole-cut interference in the region o f A2 > 1 so that the cross-section dip should be modified or be removed altogether by the above prescriptions. One final point concerns the effective trajectory defined by ( s - m 2 ) 2 d o / d t ~x s2aeff. Experimentally this IS observed to be zero or slightly negative. We obtain this result for A ~< 0,2 and 0.6 ~< A ~< 1.0. However, in the regxon near A -~ 0.25 b o t h the mixed and strong models yield aeff-~ 0.16. This is the region where there is pole-cut interference i n f 2 l e a v l n g f 1 , with its smaller cuts, to determine the energy dependence of the cross section. This does not happen m the two models discussed previously due to the altered cut phase and structure. This is a further indication that higherorder cut corrections are needed for a detailed fit to K÷A and K+Z ° photoproduction using the absorption model. Finally, we present here a summary of the various predictions for the K ÷ photoproduction observables contained in sect 2. From the charged pion photoproductlon data we predict: (i) A sharp rise in Z for 7P ~ K ÷A with a maximum near 1 at t = - m 2 while in 7P ~ K÷Z ° will be large but without a peak at t = - m 2K. (li) The recoil baryon polarization and polarized target asymmetry will be negative for 7p ~ K÷A and positive for 7 p ~ K÷~ °. With the additional constraints of exchange degeneracy we predict: (0 Z will have a dip when aK** = - 1 and the dip will be more pronounced in 7P ~ K +A than In 7P -~ K +Z°. (ii) A x and A z will both have zeroes near A2 = 1.0. (iii) A x ~- A y for small A2 for both reactmns. (Iv) Ay(O °) ~- 0 for both reactions.
216
G.R. Goldstem et al., K + photoproductzon
(v) IAvl will have a maximum at small A and will fall off with increasing A for both reac'tions. In conclusion, we have presented predictions for the general features of the observables for K+ photoproductlon. We have compared these expectations with two forms of the absorption model and pointed out several problems associated with this model. Comparisons with earlier models are also given.
References [l I [2] [3] [4] [4]
[6l [7] [8] [9] [ 10] [ 11 ] [ 12] [13]
[14] [15] [16] [17] [18] [19] [20l [21] [22] [23] [24] [251 [26l
A.M. Boyarskl et al., Phys. Rev. Letters 22 (1969) I 131. K. Gottfned and J.D Jackson, Nuovo Cimento 33 (1964) 309. F. Cooper, Phys. Rev. 177 (1969) 2398 M. Jacob and G.C. Wick, Ann. of Phys. 7 (1959) 404. J.D. Jackson, Rev. Mod Phys. 42 (1970) 12. P. Joos, Compilation of photoproductlon data above 1.2 GeV, DESY-HERA 70-1 (September 1970). J. Kim, Phys. Rev. Letters 19 (1967) 1079. M. Borghml et al., Phys Rev. Letters 25 (1970) 835 J.D. Jackson and C. Qulgg, Nucl. Phys. B22 (1970) 301. C. Michael and R. Odorico, Phys. Letters 34B (1971) 422. A.D. Martin, C. Michael and R.J.N. Phllhps, CERN preprint TH 1436 (197 l). A Capella and J. Tran Thanh Van, Nuovo Cimento Letters 4 (1970) 1199. A.G Clark and L Lyons, Nucl. Phys. B38 (1972) 37, W. Hoogland et al., Nucl. Phys. B21 (1970) 381; J. Mott et al., Phys. Rev. 177 (1969) 1966; W.L Yen et al., Phys. Rev. 188 (1969) 2011. R.L Thews, G.R. Goldstein and J.F. Owens IlI, Nucl Phys. B46 (1972) 557. N Barash-Schmldt, Rev. Mod. Phys. 43 (1971) No. 2, PartII. E.II. Harvey et al., Phys. Rev. Letters 7 (1971) 885. R.D. Field Jr., Phys. Letters 39B (1972) 389. G.L. Kane, F. tlenyey, D.R. Richards, M. Ross and G. Wilhamson, Phys. Rev. Letters 25 (1970) 1519 F.D. Gault, A.D. Martin and G L. Kane, Nuci. Phys. B32 (1971) 429. C.B. Chiu, Nucl. Phys. B30 (1971) 477. R.C. Worden, Nucl. Phys. B37 (1972) 253. M. Ross, F.S. Henyey and G.L. Kane, Nucl. Phys. B23 (1970) 269. A Borgese and M. Coloccl, A Regge cut model for the photoproducnon of K+A and K+E ° from hydrogen at high energies, CERN preprint Tll 1024 (May 1969). E.L. Berger and G. Fox, Phys. Rev. Letters 25 (1970) 1783 and Argonne preprmt ANL 7023. G.A. Rmgland et al., Nucl. Phys B44 (1972) 395. B J. tiartley and G.L. Kane, Phys. Letters 39B (1972) 531.
POLARIZATION PREDICTIONS FOR K + PHOTOPRODUCTION*
G.R. GOLDSTEIN, J.F. OWENS Ill and J. RUTHERFOORD Department of Physws, Tufts Umverstty, Medford, Mass. 02155
Recewed 5 June 1972 (Revised 19 October 1972)
Abstract: Using the existing data for charged plon photoproduction, some general features of the observables m K+ photoproduction are obtained. Using the constraints imposed by duality allows further predictions to be made. Two versions of the Reggelzed absorption model are presented and comparison is made with earher models.
1. Introduction The data currently available for the reactions 7p~K+A,
(1)
3' P ~ K + ~2°
(2)
at high energy are hmited to measurements of the differential cross section [ 1]. Experiments which measure either the polarized photon asymmetry or the recoil lambda polarizahon are now in progress. It is therefore of interest to consider the general features that may be expected for the observables that are to be measured. In this analysis we shall assume the existence of Regge poles and Regge cuts, although the precise form for the latter is not required. As shown in table 1 there are six Regge trajectories which may contribute to reactions (1) and (2). Even when the number of free parameters in a particular model is very limited the fit to the existing data need not necessarily be unique and the predictions for the various observables are certainly model dependent. It is nevertheless possible to gain some insight into the structure of the various K + photoproduction observables by considering the existing data for charged pion photoproduction. * Work partially supported by Atomic Energy Commission.
198
G.R. Goldstem et al., K + photoproductton
Table 1 Exchan{es m charged-plon and -kaon photoproductlon. KA (KB) denotes the strange member ofthe.P (~ = 1++ (1 +-) nonet. A ( 2 - ) and K ( 2 - - ) denote the Y = 0, I = 1 and Y = 1,I =1 members of the (unob~rved) J/'C = 2 - - nonet Amplitude
3' P ~ *r+n
3' P ~ K+A
f+
A2, O
K**, K*
f+
At,A(2 ' - )
KA, K ( 2 - - )
.t'~"
A2, a
K**, K*
f~
n,B
K,K B
Using SU(3) and tile current estimates for the appropriate d / f ratlos at the baryon vertex one may obtain information on the relative strengths of the various Regge exchanges. This results in several predictions for the general features of the K + photoproduction observables. By also including the constraints of exchange degeneracy and vector dominance we can make some additional, though weaker, statements about these observables. In sect. 2 we give expressions for the various observables and discuss the tmpllcatlons of the charged pion photoproductlon data. Amplitude relations required by duality are given and some possible effects of these relations are discussed. In sect. 3 we contrast the predictions of two forms of the Reggelzed absorption model. In sect. 4 we summarize our predictions and present some concluding comments on the various models for K + photoproduction.
2. Observables In order to discuss the observables for reactions (I) and (2) it is convenient to define s-channel helicity amplitudes with definite parity in the t-channel to leading order in s. We define the following set of s-channel helicity amplitudes f l = f l + , 0+' = J 1+,0- ,
(3)
f3 f4 = f l - , 0 - ' where fabcd is the amplitude for a photon of heliclty a striking a target proton with
199
G.R. Goldstein et aL, K ÷ photoproducnon
helicity b and producing a spin-zero meson and a nucleon with helicity d. Then we may define the following combinations +
f i = ~- ( f l -+f4) ' r2 =
(4)
of2
where the superscript + ( - ) indicates natural (unnatural) parity exchange in the tchannel. In calculating the observables we use the density-matrix formalism [2, 3]. The density matrix for a plane-polarized photon whose electric vector makes an angle ¢'r with respect to the scattering plane is
p'X= ½ [
]
-e-2i'3'
_e2ig"r
(5)
1
Now consider the helicity frame of the recoil baryon, i.e., the rest frame of the recoil baryon in which the z-axis is in the direction of the outgoing kaon and the y-axis is in the direction kv X k K (the normal to the scattering plane). In this frame the (unnormalized) recoil-baryon density matrix is given by
'ram'
(*,)-- ½ 4 ,-m 'oo"
m'"
(6)
The recoil baryon polarization in the ith direction (i = x, y, z) is given by
A, (¢v) = (p(~ov) o i) / (p(~pv)),
(7)
where o~ is the ith Pauli matrix and < >indicates a trace. Using eqs. (4) - (7) we obtain the expressions +*
A x(•.r)=-sin2*vlm
_
Lf1 f2 - f l
*
f221 /(p(gv) >'
(8a)
Ay (~p~,)= Im [f~l* f~2 + f l * f 2 - cos 2 , ~ (f~l*.f~2 - f l * f ~ ) i / < a ( , v ) ) ,
(8b)
A z (gv) = - sin 2 9v lm [ f l * f~l + f2- f2"* l/(P(9.r)),
(8c)
12+~212-[f112
(8d)
-br2 12)]. The amplitudes are normalized such that do
(~P.r) = ~
~
) <#(~P,),) •
(9)
200
G.R. Goldstem et al., K + photoproductton
The polarized photon asymmetry is gwen by
Y, = [do/dt (90 °) - do~dr ( 0 ° ) ] / [ d o / d t (90 °) + do/dt (0°)] :([q[2+[f~I2--~fl12--[f212)/([f([2+[.f;[2+[fl
[2+[f212 ).
(10)
In order to obtain predictions for the structure of the K + photoproduction observables from the charged-pion photoproduction data we must first make some assumptions concerning the helicity amplitude structure. We shall assume the existence of the evasive Regge pole exchanges hsted in table 1. Furthermore, we shall assume that there is a Regge cut associated with each Regge pole and that its structure is consistent with that given by the absorption model with non-flip elastic-scattering corrections. Thus, a cut which contributes to.f 1 will give an equal contribution to f4 up to an overall sign which is determined by the parity of the associated exchanged particle since both f l and f4 have overall hel,city flip equal to one. This means that cuts m f l and J4 contribute to either t~1 off1- but not to both, i.e. they have definite parity. However, f2 and f3 have overall helicity flip zero and two respectively so that their cuts are not simply related. Thus, a cut in.f2 a n d f 3 contributes to both f~" and f ~ and hence does not have definite parity. Moreover, the cuts in ]'2 do not vanish at t = tram s,nce absorptively generated cuts satisfy the t-channel conspiracy relation*. However, that portion of the cut which does not vanish in the forward direction has an extra factor of (In v + constant) in the denominator and will hence forth be called the "non-leading" cut contribution. Note that with this form for the helicity amplitudes only.f2 Is non-zero in the forward direction and h e n c e f ~ a n d f ~ become equal at t = tmm. The residue for the exchange of a strange meson in K+A photoproduction is related to the residue for the appropriate non-strange meson m lr+ photoproduction v/a SU (3) by /3 (A) = - / 3 (70 (3 f + d ) / x / ~ ( f + d).
(11)
Therefore, if all the d / f ratios are greater than - 1 or less than - 3 there will be no relative sign changes among the residues in going from n + to K ÷ photoproduction. Furthermore, if exchange degeneracy and duality are approxtmately correct, exchange-degenerate pairs of trajectories have approximately the same d / f ratios in their residues. Hence the ratio of the residue of an even-signature trajectory to the residue of its odd-signature partner is approxtmately the same for the exchange-degenerate pairs of non-strange exchanges in n + photoproduction as for the corresponding pairs of strange exchanges in K÷A photoproducnon (see table 1, where, for example, the p and A 2 residues in zr+ will have approximately the same ratm as the K* and K** residues in K+). This latter statement is not true for either rrphotoproductlon, where the ratm of an exchange-degenerate pair has a relative minus sign compared to 7r+ or K ÷, or for n °, where only one signature of the pair * Appendix B(e) of ref. [5] contains a discussion of the physical interpretation of this conspiracy relation.
G.R. Goldstem et at, K ÷photoproduction
201
can be exchanged (e.g. p not A2, B not n) and lsoscalar exchanges contribute as well. n + photoproduction is thus the more appropriate reaction for comparison. It should be emphasized however that d / f ratios for the vector-meson nonet, the pseudoscalar nonet, and the A t nonet are not required to be equal, so that the relative strengths o f the non-strange members of these multiplets can be qmte different from the strengths o f the strange members (e.g. the ,opn/rrpn ratio of couplings need not be the same as K * p A / K p A ) . Furthermore the large mass splitting between n and K will significantly depress the strength of K exchange m K + A photoproduction compared to n exchange in n + photoproduction (if the d/f ratios are ignored) since the n pole is much nearer the physical region for the latter reaction than the K is in the former reaction. We will state this more precisely in the following discusslon, but now simply stress that n + photoproduction alone does not suffice to predict K + A - more information on couplings and mass sphttings must be used. The ratio o f the residues appearing in K ÷ zo and K + A photoproduction is gwen by /3 ( Z °) //3 (A) = x / 3 ( f - d ) / ( 3 f + d).
(12)
Eq. (12) shows that there will be a sign change for the residues ifd/fis greater than 1 or less than - 3 . With the foregoing definitions we can now discuss the imphcations of the n ÷ photoproduction observables.
2.1. Polarized photon asymmetry From eq. (1 O) it can be seen that Z measures the relative amounts of naturaland unnatural-parity exchanges. Z for 7r+ photoproduction is characterized by a rapid rise to Z = + 1 at t = -mTr2 and then a gradual fall-off to Z "" 0.7 for t "~ - 1.0 GeV 2 [6]. Now, for small t, Z is dominated by f~ since both fll and f i - go to zero as ~/"S}-while f~2 and/~- get contributions from f2 which need not vamsh provided that the conspiracy relation is satisfied. Since Y = + 1 this then ~mpiles that fl~- = 0 at t = - m~r. 2 One method for achieving this is pole-cut interference where the unnatural-parity portion of the n cut cancels the unnatural-parity n pole leaving the natural-parity portion of the cut. Now consider K÷A photoproductlon where the counterpart of n exchange is K exchange. Usingd/f= 1.5 17], eq. (10) shows that the n and K residues should be about equal in magnitude. However, the K has a larger mass than the n so its contribution to the amplitudes should be approximately reduced by m2/m 2 at t = O. In addition, the point at which f~- vanishes should now be near t = -ml~. We thus expect to see )2 large and peaked near t = - m 2 for reaction ( I ) although it may not equal 1 due to the influence o f other terms which are s~zeable near t . . . . m2K. For reaction (2) we expect a somewhat different behavior. Eq. (11) shows that K exchange is suppressed in K ÷ Z ° photoproductlon using the same d/f ratio as above. Thus, the peaking near t = --m2Kwill not be present and Z will rise more smoothly
202
G.R. Goldstem et al.. K+photoproductton
to a large positive value. Th~s latter pomt follows since the natural parity K* and K** exchange are now dominant just as A 2 and # exchange would dominate n + photoproductlon if n exchange were suppressed.
2. 2. Polarized target asymmetry The polarized target asymmetry, Ty. for a target polarized perpendicularly to the scattering plane has been measured for n ÷ photoproductlon using unpolanzed photons [8] and has been found to be negative. The expression for T v is given by eq. (8b) with the opposlte sign for the f i - terms. Assuming for the moment that f i - is small then Tyis a measure of the interference between f~l and f~2. This interference results primarly from the A 2 term in f~l and the natural-panty parts of the n and A 2 cuts inf,2. Thus, the relative sign between the rr and A 2 residues is determined [9]. The d/f ratios for K** exchange have been determined in refs. [10, 11]. For the non-flip nucleon vertex d/f~- - ½ and for the flip vertex d/f~- 3.5. Thus, the K and K** d / f ratios are all greater than - 1 and hence the relative K - K * * residue sign in reaction (1) is the same as the relative 7r-A 2 residue sign in 7r+ photoproductlon. Therefore, Ty should be negative in K÷A photoproduction. The situation is different however for K ÷ y o photoproduction since now the flip K** and K couplings change sign while the non-flip K** does not. Therefore, there should be an overall sign change for Ty in reaction (2). Ty has been measured for reactions (1) and (2) but without distinguishing between the A and ~o [8]. Since the K÷A cross section ~s larger than that for K+~ ° it is reasonable to suppose that this measurement of Ty is dominated by reaction (1). The measured Ty is negatwe which is consistent with our expectations. Furthermore, if f i- is small, as will be discussed below, then Ay ~- Ty so that Ay will also be negative. Having obtained predictions for the general features of ~ and Ty we shall now consider the effects of exchange degeneracy and vector dominance in order to determine whether or not there are any further distinctive features that may be expected for the observables in reactions (1) and (2). Consider the reactions K-p ~ V° A V ° p - , K+A,
(13) V° = p , ~ , ¢,
(14)
which are related by line reversal. As noted in ref. [12] the quark duality diagram * This means that the pole terms for K - p --, p°A and K-p ---*toA will yield a purely-real phase while K - p ~ $A will have a phase e-tna. We have used vector dominance to predict these three differential cross sections from the fit to "yP--, K+A. The resulnng CA cross section is larger than the ,o°A and wA cross sections in disagreement with experiment [ 13 ]. This same disagreement is seen in other pairs of line-reversed reactions such as KN -, nA and ~rN ~ KA where the absorption model predicts the reaction with the rotating phase to be larger than the reacUon with the purely-real phase [ 14].
203
G.R. Goldstein et al., K + photoproduction
for K - p -~ p°A is an improper one and hence the imaginary part of the amplitude for this process must vamsh in the high-energy limit. The same comment * holds for K - p ~ ~oA and for ~ p -~ K÷A. Satisfying these relations puts constraints on the K*-K**, K - K B , and K A - K ( 2 - - ) couphngs in reaction (14). Vector dominance for p, ~ , and ~0 then allows us to combine the results from these three reactions to obtain a set of constraints for the amphtudes m K ÷ photoproduction. The resulting relations are Imf(K*) = ~ Imf(K**),
(15a)
l m f ( K B ) = ~- l m f ( K ) ,
(15b)
lmf(K (2--)) =] Imf(KA).
(15c)
Furthermore those relations imply that the trajectories are pairwise degenerate for K * - K * * , K B - K , and K ( 2 - - ) - K A. The latter pair of trajectories have states with jPC = 0 - - , i +÷, 2 - - , . . . . Of these the first is forbidden in a quark model m which mesons are constructed from q~l pairs. To date no 0 - - particle has been observed [ ! 5] so the exchange-degenerate K A K ( 2 - - ) residue must have a zero at a = 0. This, however, is not the case for the K - K B exchange-degenerate residue since the K pole is present at a = 0. Therefore, we expect the K A - K ( 2 - - ) terms to be suppressed relatwe to the K - K B terms by a factor o f a . The main features of K ÷ photoproduction will then be determined by the K * - K * * and K - K B trajectories. The exchange-degeneracy requirements (15) and the absence of poles in the physical region can be used to mfer the existence of nonsense wrong-signature zeroes (NWSZ) when cut effects are not present. These will occur at a = 0, - 2 , - 4 for the K * ; a = - 1 , - 3 , . . . for the K and K**; and at a = - 2 , - 4 . . . . for the K B. In the following discussion we shall assume the existence of this NWSZ structure ** 2. 3. Differential cross section
Experimentally a dip is observed in the forward direction contrary to chargedpion photoproduction. This is accounted for by the fact that the larger K mass causes the K pole to be less important in K + photoproduction as stated above. In addition, since g~N n > g 2 N x [7] the K will be even less important in K+Y:0 photoproduction. This is consistent with the data which show a more pronounced dip m the latter case. As - t increases K** exchange will become dominant. Since there is no NWSZ for K** exchange for - t < 1 this accounts for the lack of a dip at the NWSZ of the K* [121. * See footnote on facing page. ** An alternate point of view is that the entire (pole + cut) amplitude satlshes the relations (16) so that NWSZ's are not required. See sect. 3 for a more detailed discussion of such a model.
204
G.R Goldstem et al., K + photoprocluction
2•4. Polarized photon asymmetry Recent data for r r - p ~ r/n [16] indicate that there is a dip at t = --1.5 GeV 2 which could be attributed to the A 2 NWSZ. The K*--.K** trajectory lies somewhat lower than the o - A 2 trajectory so that the position of the NWSZ may be moved in shghtly. The exact location depends on the details of the K * * - P cut and how much it displaces the zero m the pole term. The differential cross section data for K ÷ photoproduction is smooth and featureless m the t region - t ~< 2.0 GeV 2 so the K*, K, and K B terms plus the various cuts must fill in the unwanted zero in the K** term. However, the vanishing of a large natural-parity contribution should have an effect on ~, causing a dip at the position of the NWSZ. Thus, measurements of Y~ in the larger t region 1.2 - 1.5 should show a dip. This dip will be more pronounced in reaction (1) which has the large unnaturalparity K-exchange contribution.
2. 5. Recoil baryon polarization From eq. (8b) we see that Ay(O Q) is a measure of the interference between fiand f~-. We expect that the terms lnfi- which are due to Kg, K ( 2 - - ) and their cuts will be small relative to the other amplitudes and hence predict Ay(O ) to be small "l'. For the case of Ay measured with unpolarlzed photons the small t region is dominated by the K** term mffl interfering with the K * * - P and K - P non-leading cut terms i n f f2. The non-zero value for the cross section at t = tmm indicates substantial values for these terms. Hence, we expect large interference effects especially near the forward direction. The phase of the non-leading cut terms is more slowly varying than that for the poles so there will be a point where the phases become nearly equal. Here IAyl will show a minimum. Thus, we expect large negative values for Ay at small t followed by a fall off to smaller negative values at larger t. From eq. (8a) we see that A x is a measure of the interference b e t w e e n f f1 and f ~ . For small t, f~- -~ff2 and hence A x ~ Ay (4~3, = 45 ° is tmphed when no other value of q~ is specified for A x or Az). As - t increases however, f~- deviates from f2 since it contains the K-pole term which has a NWSZ near t = - 1.0 GeV 2. This NWSZ will cause A x to have a zero near t = - 1.0 GeV 2. Eq. (8c) shows thatA z is dominated by the interference between ~ and f~-. Therefore, A z should also have a zero near t = - 1.0 GeV 2 due to the presence of the K NWSZ. Finally, in going from K+A to K+E ° photoproductlon the non-flip K** coupling does not change sign while the fhp K** and K couphngs do, provided that the current estrmates for the d / f ratios are approximately correct• Then f2 and f~- wdl change sign and fll will not. The Ay and A x will change sign while A z wdl not. •
O
J Recent evidence has been reported [ 17] whtch indicates possible K-K A interference in reaction (14). This would imply ]'i- ~ 0 m reactions (1) and (2). However, we expect this effect to be small for the reasons discussed previously
G.R. Goldstem et al., I ~ photoproductton
205
3. Regge cut model predictions In this section we shall consider the predictions from two different models both of which consist of Regge pole exchanges and Regge cuts generated by absorptive corrections. The two models differ in the manner m which the exchange degeneracy relations (15) are employed. In the first model it is assumed that the Regge pole amphtudes satisfy the relations (15). The residues then must have the NWSZ structure discussed in the previous section. When only elastic rescatterlng corrections are employed this model is referred to as the Argonne or "weak-cut" model. If the cut strengths are allowed to become appreciably larger than those Inferred from elastic rescattermg then the term " m t x e d " model is sometimes employed. In the usual apphcations of this model, however, it is the residue structure rather than the cut strength which differentiates it from other approaches. In the second model which we shall conslder it is assumed that if the relations (15) are satisfied then they are satisfied by the entire amplitude (pole + cut). The resldues are therefore not required to have the NWSZ structure. As In the first model the Regge cuts are generated by rescattering corrections but here It is assumed that a number of intermediate states contribute coherently thus yielding a cut strength that may be several times larger than the cut strength from elastic rescattering only. This model is usually referred to as the "strong-cut" model or SCRAM. In applying both of these models to reactions ( i ) and (2) we shall neglect K A and K(2- - ) exchanges. While these exchanges may be present we do expect that their contributions will be small as has been discussed previously. Neglecting these terms will allow us to concentrate on the major features resulting from the larger contributions of the other exchanges. Furthermore, m applying the strong cut model K B exchange will be neglected. This is consistent with the neglect of B exchange m chargedpion photoproduction 1.. The input K* and K** Regge poles for each model have the general form f l = f4 = T~l 17R vaR A ,
;2
=
%
=
_,,2 -
v °R mp+m H
R = K*,
K** ,
(16)
where v -~-l (s--u), A2 = - t + tmi n, Mp = nucleon mass, and Mi! = A or Z ° mass. 7 R and 72R are the residue functions and r/R contains the pole and the appropriate signature factor.
t For a comparison of these models in ~r° photoproductlon see ref. [ 19[. See ref. [20] for a review of Regge-cut models m general.
G R Goldstem et al.. K+ photoproduct:on
206
Table 2 Residue and signature factor structure for the pole terms in the mixed- and strong-cut models The reduced residues 3 K and 3 K are defined in the same manner as 3 K and B~ L Z 1
Residue and signature factor
Mixed model 1-e
K*
Strong-cut model K*
tlraK*
-
2 #l
K** -gl ~K**
sin nCXK,, -
K K 3" n
3'
3K (aK+ 1)
KB KB n
ha,K. *
inc~K
2
2
A +m K
26K e-~in~K
sin ~ro~K
,rd K
"-InaKB
KB 3
-'=:n~K*
K** - ~ln&K, . -2#1 e
l + e -lrraK**
l+e
t
te
ha'K* A2 +inK.2
slnnaK,
K** K** 3'1 r/
'1
(aKB+I)
l - e ~ sin ~raKB
The K and K B pole terms are gwen by
f l =f4 = 0 , 12 --13 =
R = K, K B .
The two models have different forms for the shown m table 2. The n o r m a h z a t l o n factors for chosen so that the reduced residues, 3g , are the duced residues are then assumed to be constant. The first-order cut corrections are calculated a m p h t u d e of the form
(17) residue and signature factors as the strong-cut model have been same for both models. These reusing an effective elastic-scattering
JP = ik W °t°t e --~A=Rp-
(18)
4rr t
t
whereRp= 2 a p ( l n v - ½in)+Cp. W e h a v e s e t a p = O . 4 G e V - 2 andCp
=
4.91GeV -2 by fitting the slope of the n N elastic-scattering cross secuon. We have used Oto t = 25 mb as an estimate for the initial- and final-state scattering cross sections. The resulting cuts have been enhanced by multiphcative " k " factors which take into account the lack of knowledge of the K+A and K+E ° total cross sections as well as possible interme&ate-state contributions. For the m i x e d model there are two such factors, one for the K* and K** cuts and one for the K and K B cuts "~. For the
+ We have "also obtained a fit with one k factor only m the mixed model. The resulting value was h = 2.0 and the cross-secUon fit and predictions were only shghtly changed.
G.R. Goldstein et aL, K + photoproductton
207
Table 3 The parameter values obtained for the mixed- and strong-cut models. The underhned parameters were fixed or constrained as indicated Parameter
Mixed model
Strong-cut model
01K*
0.0678
0.1025
K*
0.2293 K* 3 01
0.4527
02 K** 01 K**
0.1927
02
3 BE*
0.8506
0K
03.
0.3
0KB
0 33 ~K
K* k.~+
1.79
2.61
~*
K* ~_+
1.oo
K** M:÷ K** X+_
~K* ~K*
3.00
hK__
3.94
2 90
(d/.D+K+ *
-0.546
(d/f)K+*_
4.22
(d/]) K_
1.5
2.05
-0.236 5 73
strong-cut model there are five ~, factors, one each for the K* and K** non-flip and flip baryon vertices and one of the K cut. For both models the K* and K** trajectories were taken to be equal and were constrained to pass through the two masses with the result aK,_K** = 0.37 + 0.8 t. The K - K B trajectory was given the same slope and was constrained to pass through the K mass glvlngaK_KB = - 0 . 2 + 0.8 t. The reduced K-exchange residue was calculated by extrapolating to the particle pole and comparing with the Born term. Using the valueg2NN/4n = 15 and Kim's value d / f = 1.5 [7] we obtain the value shown m table 3. As a result of the relations (15) we have taken the d / f ratios for the K* and K** to be the same and those for the K and K B to be the same. Imposing the relations (I 5) on the pole residues then leaves 6 parameters for the mixed model. There are 1 1 parameters for the strong-cut model since these relations cannot be used for the strong-cut model and since there are more ~. factors. The parameters for both models resulting from a fit to the differential cross sections for reactions (1) and (2) are shown in table 3.
G.R. Goldstein et al., K+ photoproductton
208
1.0
I
I
I
i
I
I
I
I
I
0.I
b
-o 0.01
0.0011 0.0
I
),p--
KoA
o
5
GeV/c
A
B
GeV/c
V
II
GeV/c
x
16
G e V/c
I 0.2
I
I 0.4
I tX (GeV/c)
I O. 6
I
I 0 8
I
I I. 0
Fig. 1. The differential cross section fit to "FP ~ K÷A[ 1 ] at 5 GeV/c (0), 8 GeV/c (A), 11 GeV/c (v) and 16 GeV/c (X). Both models give virtually the same fit Table 4 Fitted d/f ratios for the mixed- and strong-cut models and the resulting coNN/pNN c o u p h n g ratios and to and p fltp/non-fhp ratios
d/f
(w/p)N~
++
+
mixed
-0.546
SCRAM
-0.236
Flip/non-flip
++
+ --
p
o9
4.22
7.8
- 0 234
11.5
-0.344
5.73
4.25
-0.405
8.8
-0.845
-
-
209
G.R Goldstein et al., K + p h o t o p r o d u c t i o n
1.0
i
!
'
.
I
,
,
.
.
0.1
f
=L
b "1o
yp ~K*~.o
0.01
0.001
I
O0
.
o
5
G eWc
A
8
Ge V / c
9
II
GeWc
x
16
GeV/c
,
0.2
.
,
0.4
I
(GeV/c)
I
0.6
I
i
0.8
I0
Fig. 2. The differential cross section fit to "tP ~ K+Z ° at the same energies as in fig. 1. The sohd line Is the mixed model and the dashed line is the strong-cut model. Figs. 1 and 2 show the fits by both models to the differential cross sections for reactions (1) and (2). The data are adequately fit in the region A ~ 1.0. However, b o t h models need relatively large cut strengths to fit the data near A = 0 v,qth the result that there is a dip near A = 1.1 in both models due to excessive pole-cut interference. This dip is not present m the data and hence if either o f these models are to w o r k for A > 1.0, higher-order cut corrections will be needed. There is also a small dip at A = 0.25 at the lower energies for K+A. This is due to pole-cut inter-
210
G.R. Goldstein et al., K + p h o t o p r o d u c t i o n
1.0
i
!
i
•
w
w
•
i
(3.8
|
~
0.6
0.4
0.0
I
0.0
0,2
=
i
0.4 0.6 •', (G eV/¢)
i
i
0 8
1.0
Fig. 3. Mixed model predictions for the polarized p h o t o n asymmetry at 5.00 G e V / c for ,3ap~ K + A ( - - ) and 35) --' K + ~ ° ( . . . . ).
1.0
0.8
0.4 ~
0.2 0.0
0.0
0.2
0.4
0.6 t, (GeV/c)
0.8
1.0
Fig. 4. Strong-cut model predictions for the polarized photon asymmetryat 5.00 GeV/c for "rP-* K+A ( - - ) and ",(p--*K+~;° (---).
G.R. Goldstein et al., K+photoproduct~on
211
0.6
/
0.4
\
/
0.2
/ t.i
/
/
/
fie
0.0
hi
'5 <
t
-0.2
i
-0.4
-0.6
-0
8
i
0.0
I
0.2
i
I
I
O.4
I
0.6
i
I
0.8
l
1.0
& (Ge V/c)
Fig. 5 Mixed model predictions for A x ( - - - ) , A y ( - - ) and Az(-.-) at 5.00 GeV/c for 3'P ~ K+A. ference m the K amplitude and is a further indication that modifications to both models are necessary before they can give a truly-detailed description of the data. On the other hand, it is encouraging that the plon-nucleon coupling constant together with Kim's value for the pseudoscalar meson d / f ratio yields a reasonable value for the K residue and for the T./A ratio at small values of A. The fitted values of the vector-tensor d / f ratios are consistent w~th the various estimates given in ref. [ 10]. Table 4 shows the resulting wNN/pNN ratios as well as the p and w flip/non-flip ratios. Figs. 3 and 4 shows the mixed and strong-cut model predictions for the polarized photon asymmetries for reactions (1) and (2). Both models show the qualitative behavior discussed m sect. 2. For both models the asymmmetries show only a slight energy dependence between 5 and 16 GeV/c with the higher-energy asymmetries being slightly more positive due to the fall-off of the K and K B contributions. Figs. 5 and 6 give the mixed model predictions for Ax, Ay and A z for reactions (1) and (2) respectively. Ay shows a large maximum followed by a fall as is discussed m sect. 2. The magnitude of the maximum is reduced for 719 -* K+E ° since the K
212
G R. Goldstem et aL, K + photoproduction
ot
0.8
e) 0 . 6 w I..-
0.4 CO
0.2
0.0
-0.2 0.0
0.2
0.4 A
0.6 (Ge V/c)
Fig. 6. Mixed model predictions for A x ( - - - ) , A y ( - - ) 3'I) -* K+~ °.
O. 8
1.0
and A z ( - . - ) at 5 00 GeV/c for
amplitude is suppressed. Both A x and A z have zeroes near A = 1.0 which result, m part, from the K NWSZ. However, since these zeroes are present m both reachons there must be additional interferences present. All of the curves show only a shght dependence in going from 5 to 16 GeV/c. Figs. 7 and 8 show the strong cut model predictions for Ax, Ay and A z m reactions (1) and (2) respectwely. Here A x is essentially the same as for the mtxed model but Ay now has a zero at A = 0.4. This Is due to the different structure of f l with its stronger cuts and the lack of a NWSZ for the K* A z is large and positive as m the mixed model but there is no longer a zero near A = 1.0. Finally, A x ~- Ay for small A in all cases as expected. The d / f ratios for both models are consistent with those in refs. [ I 0, 11 ] and as a result A x and Ay change sign while Ay does not m going from K+A to K+Z ° photoproductlon. The precedmg discussion shows that both the strong and mixed models give virtually identical fits to the differential cross sections and differ only m some predictions for the polarizations. This is a result of the fact that the dominant exchange, K**, has no NWSZ in the region considered and hence is the same for both models. The different predictions for the polarizations result from the K and K* NWSZ which appear in the mixed model but not in the strong model. There are, however, very few constraints on the large parameter set in the strong model so that
G.R. Goldstein et aL, K + p h o t o p r o d u c t i o n
213
O.B
/
06
/
/
/.
0.4
u~ 0.2 W
I
n,* I-w
0.0
./
"\//w
°/
./11
I "
.~ - 0 2
-0.4
i
-0.6
0.0
0.2
0.4
m
0.6 6. (GeV/c)
Fig. 7. Strong-cut model predictions for A x ( - - - ) ,
Ay(--)
O.B
and Az(-
0
-)
at 5 00 GeV/c f o r
~,p ~ K + A .
these predictions are not unique. As experimental data becomes available the flexibility of this model could result in different fits to the polarization asymmetries while keeping the same differential cross section fits. One further comment can be made concerning the various cut strengths obtained for both models. ;~ = 1.5 corresponds to complete absorption of the lowest pamal waves. Thus each model has cut strengths that correspond to some over-absorption. However, as has been pointed out in refs. [2 l, 22], it Is possible to obtain sundar results without over-absorption if a sharper cut-off is used for the absorbing amplitude. Therefore, over-absorption In these models need not be considered unphysical, but rather can be taken as an indication of the need for a somewhat different absorption profile.
214
G.R. Goldstem et al.. K + photoproduction
o8t 0.6
•
f
-
\
0.4 ¢Y)
I-. tad
0.2
).-
\\N
""
~ \ \ \
~
,
//
oo
/
\X
1,4 - 0.,2
-0.4
-0.6 o o
i
~
2
,
i
o 4
i
~o~
i
Fig. 8. Strong-cut model pre&ctlons for A x ( - - - ) , A y ( - - ) yp --, K+zo.
i
i
08
i
1.0
and A z ( - . - ) at 5.00 GeV/c for
4. Comments and conclusions There are two additional Regge-cut models for K+A and K÷E ° photoproductlon whose predictions may be compared with those presented in this paper. Borgese and Colocci [23] have presented a model consisting of K* exchange and a K*pomeron cut. They obtam good fits to the cross-section data and gwe predictions for the polarized photon asymmetry. Their model predicts a dip in Y, when a K , = 0 smce then only the cut terms are present and hence the natural-parity contribution has been decreased relative to the unnatural-parity part. Capella and Tran Thanh Van have proposed a model [12] with equal amounts of K* and K** exchange as well as the corresponding cut. Because the K* and K** residues are equal the NWSZ's do not lead to dips in the cross section and they obtam a good fit to all the cross section data. Furthermore, since they have chosen a phase for the cut which ~s different from that m the absorption model + the d~p at ~"The cut m ref. [ 12] Is a fixed cut whereas the absorption model gives a moving cut, l.e the phase is a function of t
G.R. Goldstem et al.. K+ photoproductton
215
A > 1 does not occur. Their predictions for E are similar to ours except that a dip at a K ** = - 1, which might be expected if the relations (15) were used, is not seen. The prediction for A y does not decrease as sharply as ours; a consequence of the different cut phase. Furthermore, they predict a zero at A2 = + 1.2. Berger and Fox [24] have also considered various model predictions for the recoil baryon polarization in reaction (2). However, their predictions are for doublecorrelation experiments where e~ther the photon or the target proton is polarized and the recoil-baryon polarization is measured. Thus, their predictions cannot be compared with ours. As indicated in the prewous section, some modifications to the cut will be required in order to remove the dip in the differential cross section for A 2 > 1. Recently, several papers have appeared which propose alterations to the cut phase and/or form in order to account for observed discrepancies in pseudoscalar mesonbaryon scattering [ 14, 25, 26]. In each case the major result is that the cuts are rotated counter-clockwise in the complex plane. The predictions for the general behavior of the A~ gwen here are not sensitive to this phase and hence are unchanged. However, the detailed shape of the A~ wdl of course be quite dependent on such details and will be different for each proposed modification. Furthermore, such cut rotations should decrease the pole-cut interference in the region o f A2 > 1 so that the cross-section dip should be modified or be removed altogether by the above prescriptions. One final point concerns the effective trajectory defined by ( s - m 2 ) 2 d o / d t ~x s2aeff. Experimentally this IS observed to be zero or slightly negative. We obtain this result for A ~< 0,2 and 0.6 ~< A ~< 1.0. However, in the regxon near A -~ 0.25 b o t h the mixed and strong models yield aeff-~ 0.16. This is the region where there is pole-cut interference i n f 2 l e a v l n g f 1 , with its smaller cuts, to determine the energy dependence of the cross section. This does not happen m the two models discussed previously due to the altered cut phase and structure. This is a further indication that higherorder cut corrections are needed for a detailed fit to K÷A and K+Z ° photoproduction using the absorption model. Finally, we present here a summary of the various predictions for the K ÷ photoproduction observables contained in sect 2. From the charged pion photoproductlon data we predict: (i) A sharp rise in Z for 7P ~ K ÷A with a maximum near 1 at t = - m 2 while in 7P ~ K÷Z ° will be large but without a peak at t = - m 2K. (li) The recoil baryon polarization and polarized target asymmetry will be negative for 7p ~ K÷A and positive for 7 p ~ K÷~ °. With the additional constraints of exchange degeneracy we predict: (0 Z will have a dip when aK** = - 1 and the dip will be more pronounced in 7P ~ K +A than In 7P -~ K +Z°. (ii) A x and A z will both have zeroes near A2 = 1.0. (iii) A x ~- A y for small A2 for both reactmns. (Iv) Ay(O °) ~- 0 for both reactions.
216
G.R. Goldstem et al., K + photoproductzon
(v) IAvl will have a maximum at small A and will fall off with increasing A for both reac'tions. In conclusion, we have presented predictions for the general features of the observables for K+ photoproductlon. We have compared these expectations with two forms of the absorption model and pointed out several problems associated with this model. Comparisons with earlier models are also given.
References [l I [2] [3] [4] [4]
[6l [7] [8] [9] [ 10] [ 11 ] [ 12] [13]
[14] [15] [16] [17] [18] [19] [20l [21] [22] [23] [24] [251 [26l
A.M. Boyarskl et al., Phys. Rev. Letters 22 (1969) I 131. K. Gottfned and J.D Jackson, Nuovo Cimento 33 (1964) 309. F. Cooper, Phys. Rev. 177 (1969) 2398 M. Jacob and G.C. Wick, Ann. of Phys. 7 (1959) 404. J.D. Jackson, Rev. Mod Phys. 42 (1970) 12. P. Joos, Compilation of photoproductlon data above 1.2 GeV, DESY-HERA 70-1 (September 1970). J. Kim, Phys. Rev. Letters 19 (1967) 1079. M. Borghml et al., Phys Rev. Letters 25 (1970) 835 J.D. Jackson and C. Qulgg, Nucl. Phys. B22 (1970) 301. C. Michael and R. Odorico, Phys. Letters 34B (1971) 422. A.D. Martin, C. Michael and R.J.N. Phllhps, CERN preprint TH 1436 (197 l). A Capella and J. Tran Thanh Van, Nuovo Cimento Letters 4 (1970) 1199. A.G Clark and L Lyons, Nucl. Phys. B38 (1972) 37, W. Hoogland et al., Nucl. Phys. B21 (1970) 381; J. Mott et al., Phys. Rev. 177 (1969) 1966; W.L Yen et al., Phys. Rev. 188 (1969) 2011. R.L Thews, G.R. Goldstein and J.F. Owens IlI, Nucl Phys. B46 (1972) 557. N Barash-Schmldt, Rev. Mod. Phys. 43 (1971) No. 2, PartII. E.II. Harvey et al., Phys. Rev. Letters 7 (1971) 885. R.D. Field Jr., Phys. Letters 39B (1972) 389. G.L. Kane, F. tlenyey, D.R. Richards, M. Ross and G. Wilhamson, Phys. Rev. Letters 25 (1970) 1519 F.D. Gault, A.D. Martin and G L. Kane, Nuci. Phys. B32 (1971) 429. C.B. Chiu, Nucl. Phys. B30 (1971) 477. R.C. Worden, Nucl. Phys. B37 (1972) 253. M. Ross, F.S. Henyey and G.L. Kane, Nucl. Phys. B23 (1970) 269. A Borgese and M. Coloccl, A Regge cut model for the photoproducnon of K+A and K+E ° from hydrogen at high energies, CERN preprint Tll 1024 (May 1969). E.L. Berger and G. Fox, Phys. Rev. Letters 25 (1970) 1783 and Argonne preprmt ANL 7023. G.A. Rmgland et al., Nucl. Phys B44 (1972) 395. B J. tiartley and G.L. Kane, Phys. Letters 39B (1972) 531.