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Regge cut contributions to exotic boson exchange cross sections
Nuclear
Physics
B34
(1971)
77-91.
North-Holland
Publishing
Company
REGGE CUT CONTRIBUTIONS TO EXOTIC BOSON EXCHANGE CROSS SECTIONS* C. QUIGG ** Institute
for Theoretical Physics, Stony Brook, Lo@ Island,
(Revised
Received manuscript
State University of New York, Neu York 11790, USA
1 June 1971 received 17 July 1971)
Abstract: Explicit calculations of near-forward differential cross sections for the single-particle-exchange forbidden reactions KNKE, K-pa%-, and are reported. The predictions are based upon an absorptive prescripn-P - K%Characteristic properties of tion for Regge cuts which satisfies s-u crossing. Regge-cut models are identified and suggestions are made for experimental studies
1. INTRODUCTION For several years it has been recognized that whereas Regge-Regge cuts may contribute significantly to scattering amplitudes for allowed processes at high energies it is easier to determine the properties of twoReggeon exchange contributions when they occur in isolation. Thus it was suggested long ago by Phillips111 that Regge-Regge cuts be studied in those ‘forbidden’ reactions such as KN - KZ which, in the absence of exotic Regge trajectories, can proceed only by multiple exchanges. Since the original suggestion, calculations have been outlined [2] and some numerical estimates of forbidden cross sections have been made by Michael [3] and others which have been useful as experimental guides. Recently more [4-6]***, sensitive experiments have reported non-zero values (rather than upper limits) for several reactions in the 3-5 GeV/c momentum range [‘I, 81 which are in rough- agreement with Michael’s expectations. In view of the rapidly improving quality of experiments and the prospect of experimental study of Regge-Regge cuts, it is appropriate to produce more detailed predictions for forbidden reactions. In this paper calculations of near-forward differential cross sections are presented for the reactions * Work supported in part under Contract No. AT(30-l)-3668B of the United States Atomic Energy Commission. ** The author holds a guest appointment at Brookhaven National Laboratory. *** Estimates of cross sections in a rescattering quark model were given by Dean 161.
C. Quigg, Kcgge cut contributions K-p - K+E- ,
(1)
K-P _ KoEo , @‘p _ K+EO , Eon + K+‘from 2 to 5 GeV/c
,
and for the reactions K-p--n
+s- ,
T-P -K+C-
(5)
,
from 2 to 16 GeV/c. The calculations reported here differ in three respects from those published previously. First, the model used for Regge cuts is one which manifestly satisfies S-U crossing [9, lo]. Second, predictions are given for the production angular distribution at small angles, not merely for do/dt at t=O. Finally, the energy dependence of the cross sections is determined by calculation at several energies, and not by estimating the location of the branch point. The paper is organized as follows. In sect. 2 the model is described and the parametrization of the single-meson exchange allowed reactions is explained. The calculation of the cross section for IcpK%-, which is technically the simplest reaction is discussed in some detail in sect. 3, where all the other results are presented as well. Some remarks on the experimental study of forbidden reactions appear in sect. 4.
2. DETAILS OF THE MODEL 2.1. Crossing symmetric model for Regge cuts The model employed for the calculation of Regge cuts is a simple extension of the usual absorptive model prescription, and has been discussed fully elsewhere [lo]. The usual model fails to satisfy S-U crossing, or what is known in Regge theory as line-reversal. That is, in general an amplitude calculated in the s-channel and crossed to the u-channel is not equal to the corresponding amplitude computed in the u-channel directly. One way to circumvent the difficulty (which is a practical problem for the study of reactions related by line-reversal) is to average the two estimates of an amplitude, and it is this recipe which I have adopted. The resulting swith normalization which corresponds to
ef:ab h J(1) + L’(12J(1)
cd:ej‘+ ,l
J(2)
Cf:Cb
J(1)
cd: eJ* h
ef:ah
J(2)
iid: e.f ,~
cid: Zj.+ ,l
J(1)
‘*J(2)
J(2)
where cl, b,. . . ,f‘label helicities as well as particle identities. Here Hs(i) is the contribution of Regge pole ‘i’ to the s-channel helicity amplitude, and Iz~(~) is its partial-wave projection which is given by il
J(i)
cdxzb = s’d(cos
cos “s)d;_/>
Bs)lfs(i)cd:ah(s,
-1
c_d(Os) . 7
The action of the operator L? in eq. (7) is to (i) sum the helicity partialwave series, (ii) line-reverse the full helicity amplitude, and (iii) reproject the desired partial wave in the new direct channel. For forbidden processes, the single-Reggeon-exchange terms Hs(i) do not contribute to (7). 2.2. Parametrizatio?is ofRegge-pole amplitudes To parametrize the single-meson exchange allowed reactions the usual decomposition into invariant amplitudes :\4= U2 fi - $iy * (ql+ q2)B)ul
,
we make
(9)
where yl(~2) is the c.m. four momentum of the incoming (outgoing) meson. The s-channel helicity amplitudes are then ,+s:+ = (Av_ + B[v+($ If-
1-t
S
where ml(m2)
-t(ml
+ m2)17_]) cosSfiS ,
= @~++B[~_(s)i-~(ml+m2)~+]}sin~OS
is the mass of the incoming +
l (El
(11)
,
(outgoing) baryon,
rn21i 1l
17, = P1+ ml1 [“2 +
(10)
and we define
PlP2 + ml)(E2
+m2)
t
7
(12)
in which /I and E are respectively the baryon momentum and energy in the center of mass system. We define the quantities rh’zA which are related to the t-channel helicity amplitudes by factors that remove the kinematical singularities: T +‘+ =A’[(ml+m2)2-t]d
,
(13)
a0
C. Quigg,
T +:- = VB
withv
=$(s-u),
A’=A+xB, x= --
(m1+9)
(ml+m2)2-t For each reaction, parameters. Thus
T
+:*
Regge
cut contributions
{-t/[(ml+m2)2t]}+ )
and (ml - m2)(n12 - n22)
Iv+
the amplitudes
2(ml +m2)
(15)
t ’
Th’:’ are characterized
by several
[ 7 + exp (-ina(t ( v ) o(t) ut * e (C +D*t) = - 2r( 1 + o(t)) sin ncu(t) y0
is the contribution of the pole with signature r. The parameters enumerated when specific computations are discussed below.
(16) will be
3. RESULTS 3.1. An As a discuss reaction that for
exchange degenerate model for the reaction Icp - K+Srepresentative example of the calculations reported here I shall in some detail the evaluation of the near-forward cross section for (1)) assumed to proceed by (K*, K**) exchange. I hasten to add reaction (1) the dominant mechanism is apparently baryon (Y= 0, I= 0, 1) exchange. The present calculation is thus an attempt to estimate the magnitude and shape of the contribution at small t generated by two successive non-exotic exchanges, and not an attempt to fit the cross section over the full angular range. A Priori, there are contributions from the K*-K*, K**-K**, and K*-K** cuts. If for simplicity we confine our attention to n”Yo intermediate states, the amplitude for the exchange of two Reggeons can be represented by the graphs in fig. 1. Upon untwisting the crossed graphs [lo], we find the resulting amplitude to be (1 + 71~2) times the contribution of the box graphs alone. Consequently the contribution of the K*-K** cut vanishes and we are left with only the even signature K*-K* and K**-K** cuts [2]. For simplicity, and to facilitate the discussion of qualitative features, we take the amplitudes fm the allowed associated production reactions K-P - nOYo from a Regge-pole fit to the available high-energy data. The amplitudes for n”Yo - K%’ are obtained by SU(3) rotations. We neglect intermediate states in which n replaces ~0, for lack of useful data on Icp -qYo, although these states in principle contribute. We then have
C. Quigg, Regge
Fig.
1. The set of graphs
Parameters
relevant
which proceed
EXD model.
C+ ~_
~.~._
45.6
2.64
ffp
Kz:,
Table 1 exchange in the best-fit
a (GeVv2)
Reaction K-p-
for the reactions KN (K*, K**) exchange.
for ~*(890)
81
cutcontributions
K-p -
n”l\
0.254
-47.3
a”A -
K+:-
2.64
-57.0
,r°Co _ K+z-
2.64
- 6.7
by
c-73.5 -59.49 42.4 -73.1
cu(0) = 0.35
1000
T
I
I,
1
II
I
P,ab- 5 GeV/c
too
l
>
IO
: \
d
a
; : 0
0. I
0.01 -1.
Fig. 2. Differential the text, computed
fGeV/c)’
cross sections for the (K*, K**) exchange reactions discussed in from the EXD fit to high-energy associated production, at 5 GeV/c incident momentum.
42
C. Quigg,
lirgge
where the individual reactions
crtl conlvibutio?is
are denoted by I:
K-p - rot1 , K-p - x0X0 ,
II: III : IV:
nO,j -, K+E- , #x:O _, K+E-
The parametrization of the amplitudes is shown in eq. (16). We require the K* and K** trajectories to be degenerate, and fix the slope of the trajectory cy’ = 0.9 GeVe2 and the scale factor v. = l/a’. For this model we choose constant residues. There remain as parameters the intercept Q(O), the vertex exponential a, and two coupling constants C+ and C-. As the intercept must be the same for all reactions we have-a total of_seven free pa. rameters to fit the reactions TN + KA, ~TN-KKS, KN --11, KN -prX, when the constraints of factorization and isospin invariance are imposed. The fits to associated production will be discussed in detail elsewhere [ 113. It suffices, for present purposes, to know that the EXD model yields a fair overall fit. The best fit parameters are given in table 1, together with the coupling constants for n”Yo -1 K%- obtained from them by SU(3) rotations. 100
L I
I
I
I K-p 5
0
I
I
I
I
,
-K’EGe
V/c
0.5
1.0
- 1. (GeV/d2
Fig. :s. Predicted cross section for K-p’K+xat 5 GeV/c. ‘The full calculation is represented by the thick, solid line. The components from K*-K” graphs, K**-K** graphs, noh intermediate states. and 7 Ox0 intermediate states are also shown scparately.
C. Quigg,
Regge
cut
83
contributions
We chose the vertex exponentials for YE equal to the one for PC, which was much better determined than the one for PA. The differential cross sections computed (from EKD poles alone) from these parameters at 5 GeV/c are plotted in fig. 2. (The reaction Icp - n-C+ is plotted rather than Icp - n°Co, for clarity.) Details, such as the forward dip for n°Co - K%‘, should not be taken too seriously, as the couplings for Icp - noA are quite uncertain. However the magnitude of the cross sections is probably reliably estimated by our simple model. The presence of the nonsense, wrong-signature zero in the K* contribution suggests that double K* exchange will be unimportant compared with double K** exchange. We will see below that this is indeed the case. In fig. 3 I have displayed the results of the calculation of Icp - KfE’ with the pole-pole cut. The thick, solid line marked ‘All’ is the full cross section implied by (17). The contribution from graphs with a So intermediate state is negligible. This suppression is a consequence of the forward dip in n°Co - K%-. The graphs with 12’ intermediate state (A) contribute nearly the entire cross section. Similarly, the K**-K** graphs (K**) are responsible for most of the cross section, and the K*-K* graphs (K*) are of little importance. To summarize the content of fig. 3, we may remark that for the model based on (17) K**- K** exchange in the two-step process Icp - noA - K+Z’ is the dominant mechanism for the peripheral (small t) peakinKp--5:‘. The K** dominance in an expected, qualitative feature, whereas the unimportance of the Co intermediate state is model-dependent. The calculated near-forward cross section for hab = 2,3, and 5 GeV/c IOOOl
I
I
I
K‘ p -
EXD
-1
Fig.
4. Predicted
I
I
I
K’r”‘ Model
(G&/c)
differential cross sections for K-p-K+% at 2, 3, 5 GeV/c EXD model for associated production.
in the
C. Quigg,
84
Regge cut contributions
is shown in fig. 4. The cross section is quite small: du/dt(t= 0) = 605, 166, 32 nb/GeV2 at &ab = 2,3,5 GeV/c. These are about an order of magnitude less than the value of 2nb/GeV2 at 3.5 GeV/c estimated by Rivers [4] or the estimate of 2.6 nb/GeV2 at 3.4 GeV/c deduced in a rescattering quark model by Dean [6]. The apparent absence in the data of forward peripheral peaks [ 121 agrees with our prediction of small cross sections, but a quantitative test is lacking. 3.2. The reacfions Kp - K”Eo, ?p - K+E’, Eon - K%It is straightforward to extend the model for Rp -K+Z’ to other reactions in the class KN -Kf. The calculations are technically more involved, as the K*-K** cut survives in these cases, so no detailed description will be given. The amplitudes for the several reactions are again represented by the graphs in fig. 1, wherein only the nY intermediate states are retained. All the required couplings may be computed from those in table 1 by isospin rotations. Results are given in fig. 5 for the reaction K”p -K%‘, and in fig. 6 for the reactions Kp - K”Eo and Eon - KE-, at momenta of 2, 3, and 5 GeV/c. It is interesting to observe that the ratio of the forward cross sections
R =g
(Kp
g
- K”Eo)
(K-p
-
K+E-)
= 20
/
in the explicit model presented here, whereas if the reactions proceeded by exchange of a single I = 1 exotic trajectory the ratio would be g. How this 20
5 _
I
EXD
I R~--DK’E~
I
I
Model
0.4 -t Fig.
5.
Predicted
05
, GEWC)
differential cross sections for lTop -KSEXD model for associated production.
at 2, 3, 5 GeV/c
in the
Regge cut contributions
C.Quigg, L”
I K-p-.m
I
I or
85
I i?”
--K’B-
;h
01
0
a2 -1
03 (GeV/d
.
04
05
Fig. 6. Predicted differential cross sections for K-p - K”p and Ron + K%5 GeV/c in the EXD model for associated production.
may come about is easy to understand. Icp-KY as (1 + 7172) [ (Kp
If we represent
at 2, 3,
the amplitude for
- noh - K+E’) + (Icp - nozo - K+E:-)] ,
where (A -B -C) represents the contribution of a box graph, and Ti is the signature of Regge pole Ii’, then the amplitude for Rp - K”Eo is -(1+27172)(K-p
-7rolI
- K+z-) + (5 + 47172)(K-p
Unless there are substantial cancellations quite plausible that the ratio R >> 1.
+ n°Co - K+z-)
among the various terms,
. it is
3.3. Exchange degenerate model with quurk model couplings To explore the sensitivity of our results to parameters and in preparation for the calculation of cross sections for Kp - n?Z’ and n-p -I&S-
Parameters 0 (GeVm2)
C+
0.658
30.095 ._
-
Table 2 for p exchange in n-p - ran.
.-.
-. .-.
D+ ._ 150.41 ._
-
._
C280.93
-~-
D-264.42
86
C. Quigg,
Regge cut contributions
EXLJ
Quark
Model
0.5 -
o.20
Fig.
7. Predicted
differential
I 0.1
1 02 -t .
I a3 Gtv/c)’
sections for K-p-+K+z:EXD quark model.
cross
EXD
Fig.
8. Predicted
differential
I 04
05
at 2, 3, .j GeV/c
in the
OuorkModel
cross sections for RapEXD quark model.
Kf?
at 2, 3, 5 GeV/c
in the
C.Quigg,
Regge cut contributio?ls I
I
I
EXD Quark
-I
Fig. 9. Predicted
,
87
I
Model
CGeV/d
diffcrcntial cross sections for K-p’ K”‘;o and 3 Ge\:/c in the EXL) quark model.
which follow, let us consider another exchange degenerate Regge-pole model for the input amplitudes. Purely for calculational case, we assume exchange degeneracy of the PO, A2, K*, and K** trajectories with a(t) = 0.5 +0.91, and assume W(3) symmetry for residue functions with D/F = 4 as prescribed by the quark model. Parameters are determined from a fit to n-p -a non, and by W(3) rotations and exchange degeneracy for all the other reactions needed. The fit to charge exchange is acceptable, but the agreement with other reactions, such as associated production, is merely schematic. Parameter values for charge exchange are recorded in table 2. Predictions based upon this model are given in fig. 7 for Kp - K+f' , in fig. 8 for K”p - K+z’, and in fig. 9 for K-p - K”Eo and Ken -.* K%-. For the small cross section, K-p - K%-, the prediction is about a factor of 10 smaller than was obtained in the EXD model discussed before. In the other cases, the EXD quark model prediction is about a factor of 2 larger than the prediction of the EXD model. 3.4. EXD quurk model wedictions for Kp - 7;+C- and n-p - K+CThe point of invoking such a simple model as the EKD quark model is to simplify the technical details of computing cross sections for ‘exotic’ associated production reactions. These processes aye distinguished from cascade production in two wags: they are observable [7], and the model calculation involves twelve separate allowed reactions, even when only the
88
C. Quigg,
Regge
cut contributions
lowest intermediate states are retained. The diagrams representing the amplitude for ICp -rn+C’ are shown in fig. 10. The calculation proceeds as outlined above. Representative differential cross sections for Rp - n+C’ and n-p - K+C- are plotted in fig. 11. In this example, the Kp cross section is about 50% higher than the one for n-p. Because the model we are using satisfies s-u crossing, this is a real effect, not simply an anomaly caused by using one model for the s-channel another for the u-channel. On the other hand, the magnitude and sign of the effect are certainly parameter dependent. It would be inaccurate to claim o(Kp) > ~(n’p) as an unambiguous prediction.
Fig.
10. The set of graphs
relevant for the reaction K-p-n+C-, (p, AZ, K* , K**) exchange.
which
proceeds
by
IO-
-
K-p -
*+I-
- 7-p -
O~ork model
CcuplingS)
K+T-
l-
. .
I.1 -
\
.
.
4. .
\
.
\
. --__ .
.
3-
8.4
. --__
-==:::_
I 1.2
\ \
;’ -
320 4
0 Fig.
11. Predicted
I
I
OJ
a2
I
-1;
I
0.3 a4 Gev/c~=
1
05
differential cross sections for K-p +n+Cand B-p’ EXD quark model. The abscissa is t’ = t -tmin.
K+x-
in the
C. Quigg,
Regge
cut contributions
89
Most of the ‘high-energy’ data on exotic associated production are confined to a forward bin. In fig. 12 the predictions of our model are compared with the data compiled by Akerlof [?I. At the lowest energies the predictions are one or two orders of magnitude below the rapidly falling data. However the calculated value is quite consistent with the new Michigan data K+C (open circles in fig. 12). More significant than the [7,13] for n’papproximate agreement in magnitude is the close agreement between the energy dependence of the Regge-cut model and the new data. The calculated cross sections fall off approximately as
!$
(t
= 0)
cT
s2”eff2
cc S-3*2 ,
corresponding to oeff N -0.6. This effective intercept is to be compared with the endpoint of the branch cut, cr,,t(O) = 0, and with the intercept appropriate for the lowest energy data shown, czeff N -4. The data on Icp -n+C’ seem to indicate a cross section which continues to fall more rapidly than the calculated result, but the data are far from clear. The present results are consistent with Michael’s estimate [3] of g
(t=O) = 1 (:Plab)-2*5
pb/(GeV/c)2
.
If one parametrizes the differential cross section as du/dt cc ext, then the parameter X N 4-5 for all the processes studied here. In the case of the first model, in which non-flip amplitudes dominate the (allowed) input processes, the slope X is about half the slope of the input cross sections. In the EKD quark model, the flip amplitudes are dominant, so the allowed cross sections are not highly peripheral. For this case the simple estimate Xcut NNaXpOle breaks down.
---
vp-.K+Xhark
10-3 4
6
+ +
mod.1 co4lpllng8
6 IO s, (GsV/C)~
20
30
Fig. 12. Energy dependence of the forward differential cross sections for K-p- T+cand n-p - K+C- predicted in the EXD quark model, Data points are from refs. [13,15].
90
4. EXPERIMENTAL
C. Quigg,
Rcggc cut contributions
STUDY OF FORBIDDEN
REACTIONS
Even in view of the order of magnitude uncertainties associated with present models for Regge-Regge cuts, detailed calculations serve a useful end if they reveal characteristics of Regge cuts which can be investigated experimentally. The results reported above shed some new light on the magnitude, parameter sensitivity, and energy dependence of Regge cut contributions to some experimentally accessible reactions. It is interesting, though expected, that the energy dependence of the cut contribution is characterized by “eff = acut-
1
rather than by the endpoint of the cut. Our calculations, and the new data with the situation envisaged by Michael [3], on n-p -* K+S- are consistent in which at a momentum of about 5 GeV/c the Regge cut contribution begins to dominate over whatever mechanism is responsible for the low-energy data. It is important to test this idea at higher energies, because if ReggeRegge cuts are identifiable, they may provide natural explanations for lowenergy line-reversal violations in allowed reactions. Except for rather wide limits which model calculations can place upon the magnitudes of Regge-Regge cuts, the identifiable characteristic is the energy dependence implied by (18). This should influence the manner in which experiments are done. Crude upper limits are no longer of any use. In an experiment run at a number of energies it is now important to accumulate similar numbers of events at all energies, in order to gain any useful information from the high-energy points. The calculations reported here have all involved stable particles in the final state. This reflects, in addition to a bias toward calculational simplicity, my judgment that the only interesting evidence for exotic exchange comes from such reactions. The evidence in reactions in which unstable hadrons are produced can always be explained away [14]. Consequently Regge cuts, even if present in resonance production will be inextricable from other effects. I am grateful to Professor J. D. Jackson for his interest and encouragement, to Dr. G. C. Fox for assorted assistance in bringing this work to completion, and to Dr. E. L. Berger for helpful suggestions. I thank Professor C. W. Akerlof and Professor J. Kirz for instructive conversations about the data. I am happy to acknowledge the cheerful service of the computer personnel at Brookhaven National Laboratory, where most of the calculations were performed.
REFERENCES [l] R. J. ii. Phillips, Phys. Letters 24B (1967) 342. (21 C. B. Chiu and J. Finkelstein, Nuovo Cimento 59A (1969) 92.
C. Quigg,
Regge
cut contributions
91
(31 C.Michael, Phys. Letters 29B (1969) 230; Nucl. Phys. B13 (1969) 644. [4] R. J. Rivers, Nuovo Cimento 57A (1968) 174. [5] C.Vossler, Nuovo Cimento 1A (1971) 357; A. B.Kaidalov and B.M.Karnakov, Yad. Fiz. 11 (1970) 216; Engl. transl. S. J. N. P. 11 (1970) 121. (61 N. W. Dean, Nucl. Phys. B7 (1968) 311. [7] C. W. Akerlof, Double charge exchange reactions, presented at the Austin Meeting of the Division of Particles and Fields of the American Physical Society, 1970: Michigan preprint UM-HE-70-19. [8] CERN-Orsay-Paris-Stockholm Collaboration, to be published; A. Lundby. private communication. [S] C. Quigg, University of California, Berkeley thesis and Lawrence Radiation Laboratory report UCRL-20032 (1970. unpublished). [lo] C.Quigg, Nucl. Phys. B29 (1971) 67. [ll] G. C. Fox and C. Quigg, to be published. [12] P.M. Dauber et al., Phys. Rev. 179 (1969) 1622. (13) C. W. Akerlof et al., Phys. Rev. Letters 27 (1971) 539. (141 E. L. Berger Phys. Rev. Letters 23 (1969) 1139. 0. I. Dahl et al., Phys. Rev. 163 (1967) 1430; [15] For n-p -+K+C-: : J. Badier et al., Saclay preprint CEA-R3037 (1966, unpublished); for K-p -n+CB-G-L-O-R Collaboration, Phys. Rev. 152 (1966) 1148; P.M. Dauber et al,, Phys. Letters 23 (1966) 154; D. Birnbaum et al., Phys. Letters 31B (1970) 36.
Physics
B34
(1971)
77-91.
North-Holland
Publishing
Company
REGGE CUT CONTRIBUTIONS TO EXOTIC BOSON EXCHANGE CROSS SECTIONS* C. QUIGG ** Institute
for Theoretical Physics, Stony Brook, Lo@ Island,
(Revised
Received manuscript
State University of New York, Neu York 11790, USA
1 June 1971 received 17 July 1971)
Abstract: Explicit calculations of near-forward differential cross sections for the single-particle-exchange forbidden reactions KNKE, K-pa%-, and are reported. The predictions are based upon an absorptive prescripn-P - K%Characteristic properties of tion for Regge cuts which satisfies s-u crossing. Regge-cut models are identified and suggestions are made for experimental studies
1. INTRODUCTION For several years it has been recognized that whereas Regge-Regge cuts may contribute significantly to scattering amplitudes for allowed processes at high energies it is easier to determine the properties of twoReggeon exchange contributions when they occur in isolation. Thus it was suggested long ago by Phillips111 that Regge-Regge cuts be studied in those ‘forbidden’ reactions such as KN - KZ which, in the absence of exotic Regge trajectories, can proceed only by multiple exchanges. Since the original suggestion, calculations have been outlined [2] and some numerical estimates of forbidden cross sections have been made by Michael [3] and others which have been useful as experimental guides. Recently more [4-6]***, sensitive experiments have reported non-zero values (rather than upper limits) for several reactions in the 3-5 GeV/c momentum range [‘I, 81 which are in rough- agreement with Michael’s expectations. In view of the rapidly improving quality of experiments and the prospect of experimental study of Regge-Regge cuts, it is appropriate to produce more detailed predictions for forbidden reactions. In this paper calculations of near-forward differential cross sections are presented for the reactions * Work supported in part under Contract No. AT(30-l)-3668B of the United States Atomic Energy Commission. ** The author holds a guest appointment at Brookhaven National Laboratory. *** Estimates of cross sections in a rescattering quark model were given by Dean 161.
C. Quigg, Kcgge cut contributions K-p - K+E- ,
(1)
K-P _ KoEo , @‘p _ K+EO , Eon + K+‘from 2 to 5 GeV/c
,
and for the reactions K-p--n
+s- ,
T-P -K+C-
(5)
,
from 2 to 16 GeV/c. The calculations reported here differ in three respects from those published previously. First, the model used for Regge cuts is one which manifestly satisfies S-U crossing [9, lo]. Second, predictions are given for the production angular distribution at small angles, not merely for do/dt at t=O. Finally, the energy dependence of the cross sections is determined by calculation at several energies, and not by estimating the location of the branch point. The paper is organized as follows. In sect. 2 the model is described and the parametrization of the single-meson exchange allowed reactions is explained. The calculation of the cross section for IcpK%-, which is technically the simplest reaction is discussed in some detail in sect. 3, where all the other results are presented as well. Some remarks on the experimental study of forbidden reactions appear in sect. 4.
2. DETAILS OF THE MODEL 2.1. Crossing symmetric model for Regge cuts The model employed for the calculation of Regge cuts is a simple extension of the usual absorptive model prescription, and has been discussed fully elsewhere [lo]. The usual model fails to satisfy S-U crossing, or what is known in Regge theory as line-reversal. That is, in general an amplitude calculated in the s-channel and crossed to the u-channel is not equal to the corresponding amplitude computed in the u-channel directly. One way to circumvent the difficulty (which is a practical problem for the study of reactions related by line-reversal) is to average the two estimates of an amplitude, and it is this recipe which I have adopted. The resulting swith normalization which corresponds to
ef:ab h J(1) + L’(12J(1)
cd:ej‘+ ,l
J(2)
Cf:Cb
J(1)
cd: eJ* h
ef:ah
J(2)
iid: e.f ,~
cid: Zj.+ ,l
J(1)
‘*J(2)
J(2)
where cl, b,. . . ,f‘label helicities as well as particle identities. Here Hs(i) is the contribution of Regge pole ‘i’ to the s-channel helicity amplitude, and Iz~(~) is its partial-wave projection which is given by il
J(i)
cdxzb = s’d(cos
cos “s)d;_/>
Bs)lfs(i)cd:ah(s,
-1
c_d(Os) . 7
The action of the operator L? in eq. (7) is to (i) sum the helicity partialwave series, (ii) line-reverse the full helicity amplitude, and (iii) reproject the desired partial wave in the new direct channel. For forbidden processes, the single-Reggeon-exchange terms Hs(i) do not contribute to (7). 2.2. Parametrizatio?is ofRegge-pole amplitudes To parametrize the single-meson exchange allowed reactions the usual decomposition into invariant amplitudes :\4= U2 fi - $iy * (ql+ q2)B)ul
,
we make
(9)
where yl(~2) is the c.m. four momentum of the incoming (outgoing) meson. The s-channel helicity amplitudes are then ,+s:+ = (Av_ + B[v+($ If-
1-t
S
where ml(m2)
-t(ml
+ m2)17_]) cosSfiS ,
= @~++B[~_(s)i-~(ml+m2)~+]}sin~OS
is the mass of the incoming +
l (El
(11)
,
(outgoing) baryon,
rn21i 1l
17, = P1+ ml1 [“2 +
(10)
and we define
PlP2 + ml)(E2
+m2)
t
7
(12)
in which /I and E are respectively the baryon momentum and energy in the center of mass system. We define the quantities rh’zA which are related to the t-channel helicity amplitudes by factors that remove the kinematical singularities: T +‘+ =A’[(ml+m2)2-t]d
,
(13)
a0
C. Quigg,
T +:- = VB
withv
=$(s-u),
A’=A+xB, x= --
(m1+9)
(ml+m2)2-t For each reaction, parameters. Thus
T
+:*
Regge
cut contributions
{-t/[(ml+m2)2t]}+ )
and (ml - m2)(n12 - n22)
Iv+
the amplitudes
2(ml +m2)
(15)
t ’
Th’:’ are characterized
by several
[ 7 + exp (-ina(t ( v ) o(t) ut * e (C +D*t) = - 2r( 1 + o(t)) sin ncu(t) y0
is the contribution of the pole with signature r. The parameters enumerated when specific computations are discussed below.
(16) will be
3. RESULTS 3.1. An As a discuss reaction that for
exchange degenerate model for the reaction Icp - K+Srepresentative example of the calculations reported here I shall in some detail the evaluation of the near-forward cross section for (1)) assumed to proceed by (K*, K**) exchange. I hasten to add reaction (1) the dominant mechanism is apparently baryon (Y= 0, I= 0, 1) exchange. The present calculation is thus an attempt to estimate the magnitude and shape of the contribution at small t generated by two successive non-exotic exchanges, and not an attempt to fit the cross section over the full angular range. A Priori, there are contributions from the K*-K*, K**-K**, and K*-K** cuts. If for simplicity we confine our attention to n”Yo intermediate states, the amplitude for the exchange of two Reggeons can be represented by the graphs in fig. 1. Upon untwisting the crossed graphs [lo], we find the resulting amplitude to be (1 + 71~2) times the contribution of the box graphs alone. Consequently the contribution of the K*-K** cut vanishes and we are left with only the even signature K*-K* and K**-K** cuts [2]. For simplicity, and to facilitate the discussion of qualitative features, we take the amplitudes fm the allowed associated production reactions K-P - nOYo from a Regge-pole fit to the available high-energy data. The amplitudes for n”Yo - K%’ are obtained by SU(3) rotations. We neglect intermediate states in which n replaces ~0, for lack of useful data on Icp -qYo, although these states in principle contribute. We then have
C. Quigg, Regge
Fig.
1. The set of graphs
Parameters
relevant
which proceed
EXD model.
C+ ~_
~.~._
45.6
2.64
ffp
Kz:,
Table 1 exchange in the best-fit
a (GeVv2)
Reaction K-p-
for the reactions KN (K*, K**) exchange.
for ~*(890)
81
cutcontributions
K-p -
n”l\
0.254
-47.3
a”A -
K+:-
2.64
-57.0
,r°Co _ K+z-
2.64
- 6.7
by
c-73.5 -59.49 42.4 -73.1
cu(0) = 0.35
1000
T
I
I,
1
II
I
P,ab- 5 GeV/c
too
l
>
IO
: \
d
a
; : 0
0. I
0.01 -1.
Fig. 2. Differential the text, computed
fGeV/c)’
cross sections for the (K*, K**) exchange reactions discussed in from the EXD fit to high-energy associated production, at 5 GeV/c incident momentum.
42
C. Quigg,
lirgge
where the individual reactions
crtl conlvibutio?is
are denoted by I:
K-p - rot1 , K-p - x0X0 ,
II: III : IV:
nO,j -, K+E- , #x:O _, K+E-
The parametrization of the amplitudes is shown in eq. (16). We require the K* and K** trajectories to be degenerate, and fix the slope of the trajectory cy’ = 0.9 GeVe2 and the scale factor v. = l/a’. For this model we choose constant residues. There remain as parameters the intercept Q(O), the vertex exponential a, and two coupling constants C+ and C-. As the intercept must be the same for all reactions we have-a total of_seven free pa. rameters to fit the reactions TN + KA, ~TN-KKS, KN --11, KN -prX, when the constraints of factorization and isospin invariance are imposed. The fits to associated production will be discussed in detail elsewhere [ 113. It suffices, for present purposes, to know that the EXD model yields a fair overall fit. The best fit parameters are given in table 1, together with the coupling constants for n”Yo -1 K%- obtained from them by SU(3) rotations. 100
L I
I
I
I K-p 5
0
I
I
I
I
,
-K’EGe
V/c
0.5
1.0
- 1. (GeV/d2
Fig. :s. Predicted cross section for K-p’K+xat 5 GeV/c. ‘The full calculation is represented by the thick, solid line. The components from K*-K” graphs, K**-K** graphs, noh intermediate states. and 7 Ox0 intermediate states are also shown scparately.
C. Quigg,
Regge
cut
83
contributions
We chose the vertex exponentials for YE equal to the one for PC, which was much better determined than the one for PA. The differential cross sections computed (from EKD poles alone) from these parameters at 5 GeV/c are plotted in fig. 2. (The reaction Icp - n-C+ is plotted rather than Icp - n°Co, for clarity.) Details, such as the forward dip for n°Co - K%‘, should not be taken too seriously, as the couplings for Icp - noA are quite uncertain. However the magnitude of the cross sections is probably reliably estimated by our simple model. The presence of the nonsense, wrong-signature zero in the K* contribution suggests that double K* exchange will be unimportant compared with double K** exchange. We will see below that this is indeed the case. In fig. 3 I have displayed the results of the calculation of Icp - KfE’ with the pole-pole cut. The thick, solid line marked ‘All’ is the full cross section implied by (17). The contribution from graphs with a So intermediate state is negligible. This suppression is a consequence of the forward dip in n°Co - K%-. The graphs with 12’ intermediate state (A) contribute nearly the entire cross section. Similarly, the K**-K** graphs (K**) are responsible for most of the cross section, and the K*-K* graphs (K*) are of little importance. To summarize the content of fig. 3, we may remark that for the model based on (17) K**- K** exchange in the two-step process Icp - noA - K+Z’ is the dominant mechanism for the peripheral (small t) peakinKp--5:‘. The K** dominance in an expected, qualitative feature, whereas the unimportance of the Co intermediate state is model-dependent. The calculated near-forward cross section for hab = 2,3, and 5 GeV/c IOOOl
I
I
I
K‘ p -
EXD
-1
Fig.
4. Predicted
I
I
I
K’r”‘ Model
(G&/c)
differential cross sections for K-p-K+% at 2, 3, 5 GeV/c EXD model for associated production.
in the
C. Quigg,
84
Regge cut contributions
is shown in fig. 4. The cross section is quite small: du/dt(t= 0) = 605, 166, 32 nb/GeV2 at &ab = 2,3,5 GeV/c. These are about an order of magnitude less than the value of 2nb/GeV2 at 3.5 GeV/c estimated by Rivers [4] or the estimate of 2.6 nb/GeV2 at 3.4 GeV/c deduced in a rescattering quark model by Dean [6]. The apparent absence in the data of forward peripheral peaks [ 121 agrees with our prediction of small cross sections, but a quantitative test is lacking. 3.2. The reacfions Kp - K”Eo, ?p - K+E’, Eon - K%It is straightforward to extend the model for Rp -K+Z’ to other reactions in the class KN -Kf. The calculations are technically more involved, as the K*-K** cut survives in these cases, so no detailed description will be given. The amplitudes for the several reactions are again represented by the graphs in fig. 1, wherein only the nY intermediate states are retained. All the required couplings may be computed from those in table 1 by isospin rotations. Results are given in fig. 5 for the reaction K”p -K%‘, and in fig. 6 for the reactions Kp - K”Eo and Eon - KE-, at momenta of 2, 3, and 5 GeV/c. It is interesting to observe that the ratio of the forward cross sections
R =g
(Kp
g
- K”Eo)
(K-p
-
K+E-)
= 20
/
in the explicit model presented here, whereas if the reactions proceeded by exchange of a single I = 1 exotic trajectory the ratio would be g. How this 20
5 _
I
EXD
I R~--DK’E~
I
I
Model
0.4 -t Fig.
5.
Predicted
05
, GEWC)
differential cross sections for lTop -KSEXD model for associated production.
at 2, 3, 5 GeV/c
in the
Regge cut contributions
C.Quigg, L”
I K-p-.m
I
I or
85
I i?”
--K’B-
;h
01
0
a2 -1
03 (GeV/d
.
04
05
Fig. 6. Predicted differential cross sections for K-p - K”p and Ron + K%5 GeV/c in the EXD model for associated production.
may come about is easy to understand. Icp-KY as (1 + 7172) [ (Kp
If we represent
at 2, 3,
the amplitude for
- noh - K+E’) + (Icp - nozo - K+E:-)] ,
where (A -B -C) represents the contribution of a box graph, and Ti is the signature of Regge pole Ii’, then the amplitude for Rp - K”Eo is -(1+27172)(K-p
-7rolI
- K+z-) + (5 + 47172)(K-p
Unless there are substantial cancellations quite plausible that the ratio R >> 1.
+ n°Co - K+z-)
among the various terms,
. it is
3.3. Exchange degenerate model with quurk model couplings To explore the sensitivity of our results to parameters and in preparation for the calculation of cross sections for Kp - n?Z’ and n-p -I&S-
Parameters 0 (GeVm2)
C+
0.658
30.095 ._
-
Table 2 for p exchange in n-p - ran.
.-.
-. .-.
D+ ._ 150.41 ._
-
._
C280.93
-~-
D-264.42
86
C. Quigg,
Regge cut contributions
EXLJ
Quark
Model
0.5 -
o.20
Fig.
7. Predicted
differential
I 0.1
1 02 -t .
I a3 Gtv/c)’
sections for K-p-+K+z:EXD quark model.
cross
EXD
Fig.
8. Predicted
differential
I 04
05
at 2, 3, .j GeV/c
in the
OuorkModel
cross sections for RapEXD quark model.
Kf?
at 2, 3, 5 GeV/c
in the
C.Quigg,
Regge cut contributio?ls I
I
I
EXD Quark
-I
Fig. 9. Predicted
,
87
I
Model
CGeV/d
diffcrcntial cross sections for K-p’ K”‘;o and 3 Ge\:/c in the EXL) quark model.
which follow, let us consider another exchange degenerate Regge-pole model for the input amplitudes. Purely for calculational case, we assume exchange degeneracy of the PO, A2, K*, and K** trajectories with a(t) = 0.5 +0.91, and assume W(3) symmetry for residue functions with D/F = 4 as prescribed by the quark model. Parameters are determined from a fit to n-p -a non, and by W(3) rotations and exchange degeneracy for all the other reactions needed. The fit to charge exchange is acceptable, but the agreement with other reactions, such as associated production, is merely schematic. Parameter values for charge exchange are recorded in table 2. Predictions based upon this model are given in fig. 7 for Kp - K+f' , in fig. 8 for K”p - K+z’, and in fig. 9 for K-p - K”Eo and Ken -.* K%-. For the small cross section, K-p - K%-, the prediction is about a factor of 10 smaller than was obtained in the EXD model discussed before. In the other cases, the EXD quark model prediction is about a factor of 2 larger than the prediction of the EXD model. 3.4. EXD quurk model wedictions for Kp - 7;+C- and n-p - K+CThe point of invoking such a simple model as the EKD quark model is to simplify the technical details of computing cross sections for ‘exotic’ associated production reactions. These processes aye distinguished from cascade production in two wags: they are observable [7], and the model calculation involves twelve separate allowed reactions, even when only the
88
C. Quigg,
Regge
cut contributions
lowest intermediate states are retained. The diagrams representing the amplitude for ICp -rn+C’ are shown in fig. 10. The calculation proceeds as outlined above. Representative differential cross sections for Rp - n+C’ and n-p - K+C- are plotted in fig. 11. In this example, the Kp cross section is about 50% higher than the one for n-p. Because the model we are using satisfies s-u crossing, this is a real effect, not simply an anomaly caused by using one model for the s-channel another for the u-channel. On the other hand, the magnitude and sign of the effect are certainly parameter dependent. It would be inaccurate to claim o(Kp) > ~(n’p) as an unambiguous prediction.
Fig.
10. The set of graphs
relevant for the reaction K-p-n+C-, (p, AZ, K* , K**) exchange.
which
proceeds
by
IO-
-
K-p -
*+I-
- 7-p -
O~ork model
CcuplingS)
K+T-
l-
. .
I.1 -
\
.
.
4. .
\
.
\
. --__ .
.
3-
8.4
. --__
-==:::_
I 1.2
\ \
;’ -
320 4
0 Fig.
11. Predicted
I
I
OJ
a2
I
-1;
I
0.3 a4 Gev/c~=
1
05
differential cross sections for K-p +n+Cand B-p’ EXD quark model. The abscissa is t’ = t -tmin.
K+x-
in the
C. Quigg,
Regge
cut contributions
89
Most of the ‘high-energy’ data on exotic associated production are confined to a forward bin. In fig. 12 the predictions of our model are compared with the data compiled by Akerlof [?I. At the lowest energies the predictions are one or two orders of magnitude below the rapidly falling data. However the calculated value is quite consistent with the new Michigan data K+C (open circles in fig. 12). More significant than the [7,13] for n’papproximate agreement in magnitude is the close agreement between the energy dependence of the Regge-cut model and the new data. The calculated cross sections fall off approximately as
!$
(t
= 0)
cT
s2”eff2
cc S-3*2 ,
corresponding to oeff N -0.6. This effective intercept is to be compared with the endpoint of the branch cut, cr,,t(O) = 0, and with the intercept appropriate for the lowest energy data shown, czeff N -4. The data on Icp -n+C’ seem to indicate a cross section which continues to fall more rapidly than the calculated result, but the data are far from clear. The present results are consistent with Michael’s estimate [3] of g
(t=O) = 1 (:Plab)-2*5
pb/(GeV/c)2
.
If one parametrizes the differential cross section as du/dt cc ext, then the parameter X N 4-5 for all the processes studied here. In the case of the first model, in which non-flip amplitudes dominate the (allowed) input processes, the slope X is about half the slope of the input cross sections. In the EKD quark model, the flip amplitudes are dominant, so the allowed cross sections are not highly peripheral. For this case the simple estimate Xcut NNaXpOle breaks down.
---
vp-.K+Xhark
10-3 4
6
+ +
mod.1 co4lpllng8
6 IO s, (GsV/C)~
20
30
Fig. 12. Energy dependence of the forward differential cross sections for K-p- T+cand n-p - K+C- predicted in the EXD quark model, Data points are from refs. [13,15].
90
4. EXPERIMENTAL
C. Quigg,
Rcggc cut contributions
STUDY OF FORBIDDEN
REACTIONS
Even in view of the order of magnitude uncertainties associated with present models for Regge-Regge cuts, detailed calculations serve a useful end if they reveal characteristics of Regge cuts which can be investigated experimentally. The results reported above shed some new light on the magnitude, parameter sensitivity, and energy dependence of Regge cut contributions to some experimentally accessible reactions. It is interesting, though expected, that the energy dependence of the cut contribution is characterized by “eff = acut-
1
rather than by the endpoint of the cut. Our calculations, and the new data with the situation envisaged by Michael [3], on n-p -* K+S- are consistent in which at a momentum of about 5 GeV/c the Regge cut contribution begins to dominate over whatever mechanism is responsible for the low-energy data. It is important to test this idea at higher energies, because if ReggeRegge cuts are identifiable, they may provide natural explanations for lowenergy line-reversal violations in allowed reactions. Except for rather wide limits which model calculations can place upon the magnitudes of Regge-Regge cuts, the identifiable characteristic is the energy dependence implied by (18). This should influence the manner in which experiments are done. Crude upper limits are no longer of any use. In an experiment run at a number of energies it is now important to accumulate similar numbers of events at all energies, in order to gain any useful information from the high-energy points. The calculations reported here have all involved stable particles in the final state. This reflects, in addition to a bias toward calculational simplicity, my judgment that the only interesting evidence for exotic exchange comes from such reactions. The evidence in reactions in which unstable hadrons are produced can always be explained away [14]. Consequently Regge cuts, even if present in resonance production will be inextricable from other effects. I am grateful to Professor J. D. Jackson for his interest and encouragement, to Dr. G. C. Fox for assorted assistance in bringing this work to completion, and to Dr. E. L. Berger for helpful suggestions. I thank Professor C. W. Akerlof and Professor J. Kirz for instructive conversations about the data. I am happy to acknowledge the cheerful service of the computer personnel at Brookhaven National Laboratory, where most of the calculations were performed.
REFERENCES [l] R. J. ii. Phillips, Phys. Letters 24B (1967) 342. (21 C. B. Chiu and J. Finkelstein, Nuovo Cimento 59A (1969) 92.
C. Quigg,
Regge
cut contributions
91
(31 C.Michael, Phys. Letters 29B (1969) 230; Nucl. Phys. B13 (1969) 644. [4] R. J. Rivers, Nuovo Cimento 57A (1968) 174. [5] C.Vossler, Nuovo Cimento 1A (1971) 357; A. B.Kaidalov and B.M.Karnakov, Yad. Fiz. 11 (1970) 216; Engl. transl. S. J. N. P. 11 (1970) 121. (61 N. W. Dean, Nucl. Phys. B7 (1968) 311. [7] C. W. Akerlof, Double charge exchange reactions, presented at the Austin Meeting of the Division of Particles and Fields of the American Physical Society, 1970: Michigan preprint UM-HE-70-19. [8] CERN-Orsay-Paris-Stockholm Collaboration, to be published; A. Lundby. private communication. [S] C. Quigg, University of California, Berkeley thesis and Lawrence Radiation Laboratory report UCRL-20032 (1970. unpublished). [lo] C.Quigg, Nucl. Phys. B29 (1971) 67. [ll] G. C. Fox and C. Quigg, to be published. [12] P.M. Dauber et al., Phys. Rev. 179 (1969) 1622. (13) C. W. Akerlof et al., Phys. Rev. Letters 27 (1971) 539. (141 E. L. Berger Phys. Rev. Letters 23 (1969) 1139. 0. I. Dahl et al., Phys. Rev. 163 (1967) 1430; [15] For n-p -+K+C-: : J. Badier et al., Saclay preprint CEA-R3037 (1966, unpublished); for K-p -n+CB-G-L-O-R Collaboration, Phys. Rev. 152 (1966) 1148; P.M. Dauber et al,, Phys. Letters 23 (1966) 154; D. Birnbaum et al., Phys. Letters 31B (1970) 36.