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PHYSICAL REVIEW VOLUME 180, NUMBER 5 25 APRIL 1969 Forward Photoproduction of ^+ and K+ Mesons in the Regge-Pole Model* C. C. SHIH Brookhaven National Laboratory, Upton, Nmv York AND Wu-Ki TuNGf Institute for Theoretical Physics, State University of New York, Stony Brook, New York (Received 4 November 1968; revised manuscript received 19 December 1968) High-energy forward photoproduction of 7r+ and ir+ are analyzed and compared in the Regge-pole model. It is pointed out that, despite the apparent difference in the shape of the differential cross section for the two processes, existing data are consistent with the exchange of a parity doublet 7r,7rc or K,Kc as the case may be. The differences arise from the equal and unequal baryon mass kinematics involved in the two cases, respectively. A T high energies, the observed differential cross section for the photoproduction of pions, yp-^ TT+n, shows a marked peak in the forward direction.^ On the other hand, measurements on the photopro- duction of K mesons, yp —> K^Y, reveal a flat differen- tial cross section.^ In a Regge-pole model, the data on yp —» TT'^n can be fitted by the exchange of a parity doublet consisting of the pion and a conspirator tra- jectory TTc.^'^ A similar model for 7/?—>ir+F, with exchange of K and Kc trajectories, appears to fail rather badly.^-^ An ad hoc ir*-exchange contribution has to be added to reproduce the experimental data.^ The purpose of this paper is to point out that the apparent difficulties encountered in fitting the K- meson photoproduction data with K and Kc exchange come from an inadequate treatment of the kinematics near /=0. A careful parametrization of the residue functions leads to good agreement with experimental data. The difference in the observed differential cross sections for TT and i^ photoproduction can be naturally accounted for in this model by the equal- and unequal- mass kinematics involved in the two cases, respectively. In order to make the essential points clear, we shall confine our considerations strictly to the near-forward direction and use the least number of free parameters possible. Let us first consider yp —^ K^Y. The kinematic- singularity-free amplitudes are^ (1) * Work supported in part by the U. S. Atomic Energy Commis- sion. t Present address: Institute for Advanced Study, Princeton, N.J, 1 A. M. Boyarski et al., Phys. Rev. Letters 20, 300 (1968) (private communication). 2 J. S. Ball, W. R. Frazer, and M. Jacob, Phys. Rev. Letters 20, 518 (1968). ^ F. S. Henyey, Phys. Rev. 170, 1619 (1968). 4 J. P. Ader, M. Capdeville, and H. Navelet, Nuovo Cimento (to be published). Note that the amplitude /oi~ given in Ref. 2 differs from that in Eq. (1) by a factor of (t—iiK^). The same difference appears also for the corresponding amplitude in pion photoproduction below. The reason that Eq. (1) is the more logical choice is explained in W. A. Bardeen and Wu-ki Tung, Phys. Rev. 173, 1423 (1968). (Reference 3 also contains a discus- sion on this point.) (4) where f^ are the usual ^channel parity-conserving helicity amplitudes, ^K is the mass of ir+, and m±=mY zkm:ti' The kinematic constraints which are of relevance for forward scattering are^"^ /ir=(m_/^+)/ii+, att=0 (2) fn-= m-fio~, at /= mJ. (3) These two constraint equations, at very close by points (w_^c^0.04 GeV^), have important implications for what follows. We assume the exchange of a pair of trajectories aK(t) and ax,(0 satisfying ax(0) = aKc(0). Furthermore, for obvious reasons, we assume that the K trajectory chooses sense while the Kc trajectory chooses nonsense at Q;X=0 and aKc=0, respectively. Following the usual procedure of Reggeization,^ we obtain the following asymptotic expressions for the helicity amplitudes^: /io-=fWo:K(0)Tio-(V^o)«^-S s M. Gell-Mann et al, Phys. Rev. 133, B145 (1964). ^ There is some subtlety involved in the derivation of Eq. (4) that is common in the Reggeization of photon amphtudes. We give the details of the derivation here. The standard procedure for Reggeization gives /oi,ir = r(^)ax(OiSio-(V^o)«^-S /oi,i-r = r(0 XaK^(t)(3n~(s/so)''^~^. We can expand ax(0 around the point t=fiK^ [remember aR (m^) = 0], ax (t) = {t-fiK^)a'+i (t-m^)W'+ ... = (^_^^2)[Q:'_|_I(/_^^2)C^;''-J ]. Substituting these relations into Eq. (1), we see that one factor of axit) cancels the kinematic factor (t—fiE^)'^ in (1). Since /v"" are the amphtudes that are free of kinematic singularities, ^\^~ must have zeros to eliminate the remaining singularities. We therefore arrive at the first two equations in Eq. (4). On the other hand, assuming the Kc trajec- tory to choose nonsense, we have /oi,H"^ = fc(0«ifc(0/5io"*"(V'^o)"^'=~'S foi,hh^ = ^c{l'hKcQ)^ii^{s/so)''^<>~^. In general we do not know the behavior of axc near t=ixK^. [We only know aKc{^)=0LK(fi).'] Substituting these results into Eq. (1), the a factors remain intact while the /Sx^"*" factor again must ehminate the singular factors. We thus get the last two equations of Eq. (4). One may wonder whether it is really correct to cancel out one factor of aK against the kinematic factor {t—jXR^) appearing in Eq. (1) for the f\^~ amphtudes. There is no compelling reason to do so in the pure Regge formahsm. However, here the issue is decided by the requirement that at t=tJ,K^ the Regge form for the amphtude /io~ must agree with that for elementary K ex- change. Had we not cancelled one aK factor, the amphtude /oi would vanish at t=tJ^K^ in contrast to the elementary K exchange which gives a finite contribution. These same considerations also apply to pion photoproduction. 180 1446

Mesons in the Regge-Pole Model

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Page 1: Mesons in the Regge-Pole Model

P H Y S I C A L R E V I E W V O L U M E 1 8 0 , N U M B E R 5 2 5 A P R I L 1 9 6 9

Forward Photoproduction of ^+ and K+ Mesons in the Regge-Pole Model*

C. C. SHIH

Brookhaven National Laboratory, Upton, Nmv York

AND

Wu-Ki TuNGf Institute for Theoretical Physics, State University of New York, Stony Brook, New York

(Received 4 November 1968; revised manuscript received 19 December 1968)

High-energy forward photoproduction of 7r+ and ir+ are analyzed and compared in the Regge-pole model. I t is pointed out that, despite the apparent difference in the shape of the differential cross section for the two processes, existing data are consistent with the exchange of a parity doublet 7r,7rc or K,Kc as the case may be. The differences arise from the equal and unequal baryon mass kinematics involved in the two cases, respectively.

AT high energies, the observed differential cross section for the photoproduction of pions, yp-^

TT+n, shows a marked peak in the forward direction.^ On the other hand, measurements on the photopro­duction of K mesons, yp —> K^Y, reveal a flat differen­tial cross section.^ In a Regge-pole model, the data on yp —» TT'^n can be fitted by the exchange of a parity doublet consisting of the pion and a conspirator tra­jectory TTc.̂ '̂ A similar model for 7/?—>ir+F, with exchange of K and Kc trajectories, appears to fail rather badly.^-^ An ad hoc ir*-exchange contribution has to be added to reproduce the experimental data.^

The purpose of this paper is to point out that the apparent difficulties encountered in fitting the K-meson photoproduction data with K and Kc exchange come from an inadequate treatment of the kinematics near / = 0 . A careful parametrization of the residue functions leads to good agreement with experimental data. The difference in the observed differential cross sections for TT and i^ photoproduction can be naturally accounted for in this model by the equal- and unequal-mass kinematics involved in the two cases, respectively. In order to make the essential points clear, we shall confine our considerations strictly to the near-forward direction and use the least number of free parameters possible.

Let us first consider yp —^ K^Y. The kinematic-singularity-free amplitudes are^

(1)

* Work supported in part by the U. S. Atomic Energy Commis­sion.

t Present address: Institute for Advanced Study, Princeton, N.J,

1 A. M. Boyarski et al., Phys. Rev. Letters 20, 300 (1968) (private communication).

2 J. S. Ball, W. R. Frazer, and M. Jacob, Phys. Rev. Letters 20, 518 (1968).

^ F. S. Henyey, Phys. Rev. 170, 1619 (1968). 4 J. P. Ader, M. Capdeville, and H. Navelet, Nuovo Cimento

(to be published). Note that the amplitude /oi~ given in Ref. 2 differs from that in Eq. (1) by a factor of (t—iiK^). The same difference appears also for the corresponding amplitude in pion photoproduction below. The reason that Eq. (1) is the more logical choice is explained in W. A. Bardeen and Wu-ki Tung, Phys. Rev. 173, 1423 (1968). (Reference 3 also contains a discus­sion on this point.)

(4)

where f^ are the usual ^channel parity-conserving helicity amplitudes, ^K is the mass of ir+, and m±=mY zkm:ti' The kinematic constraints which are of relevance for forward scattering are^"^

/ i r = ( m _ / ^ + ) / i i + , a t t = 0 (2)

fn-= m-fio~, at / = mJ. (3)

These two constraint equations, at very close by points (w_^c^0.04 GeV^), have important implications for what follows.

We assume the exchange of a pair of trajectories aK(t) and ax, (0 satisfying ax(0) = aKc(0). Furthermore, for obvious reasons, we assume that the K trajectory chooses sense while the Kc trajectory chooses nonsense at Q;X=0 and aKc=0, respectively. Following the usual procedure of Reggeization,^ we obtain the following asymptotic expressions for the helicity amplitudes^:

/io-=fWo:K(0)Tio-(V^o)«^-S

s M. Gell-Mann et al, Phys. Rev. 133, B145 (1964). ^ There is some subtlety involved in the derivation of Eq. (4)

that is common in the Reggeization of photon amphtudes. We give the details of the derivation here. The standard procedure for Reggeization gives /oi, ir = r(^)ax(OiSio-(V^o)«^-S /oi , i - r = r (0 XaK^(t)(3n~(s/so)''^~^. We can expand ax(0 around the point t=fiK^ [remember aR (m^) = 0], ax (t) = {t-fiK^)a'+i (t-m^)W'+ . . . = (^_^^2)[Q:'_|_I(/_^^2)C^;' '-J ] . Substituting these relations into Eq. (1), we see that one factor of axit) cancels the kinematic factor (t—fiE^)'^ in (1). Since /v"" are the amphtudes that are free of kinematic singularities, ^\^~ must have zeros to eliminate the remaining singularities. We therefore arrive at the first two equations in Eq. (4). On the other hand, assuming the Kc trajec­tory to choose nonsense, we have /oi,H"^ = fc(0«ifc(0/5io"*"(V'̂ o)"̂ '=~'S foi,hh^ = ̂ c{l'hKcQ)^ii^{s/so)''^<>~^. In general we do not know the behavior of axc near t=ixK^. [We only know aKc{^)=0LK(fi).'] Substituting these results into Eq. (1), the a factors remain intact while the /Sx̂"*" factor again must ehminate the singular factors. We thus get the last two equations of Eq. (4).

One may wonder whether it is really correct to cancel out one factor of aK against the kinematic factor {t—jXR^) appearing in Eq. (1) for the f\^~ amphtudes. There is no compelling reason to do so in the pure Regge formahsm. However, here the issue is decided by the requirement that at t=tJ,K^ the Regge form for the amphtude /io~ must agree with that for elementary K ex­change. Had we not cancelled one aK factor, the amphtude /oi would vanish at t=tJ^K^ in contrast to the elementary K exchange which gives a finite contribution. These same considerations also apply to pion photoproduction.

180 1446

Page 2: Mesons in the Regge-Pole Model

180 P H O T O P R O D U C T I O N O F TT + A N D K^ 1447

where f (/)= (e~*'^"+l)/sin7rQ: is the signature factor [OLK(0) is inserted for convenience]. The reduced residue functions y\^,^ are free of kinematic singularities. However, they are not free from kinematic zeros. In particular, Eqs. (2) and (3) impose two constraints on these residue functions. We can parametrize yx^,^ in a way such that these constraints are automatically satisfied. For instance, we can write

(5)

7io-(0 = ^ i W + ^ 2 ( / ) ,

711-W = lm-/{mJ-y^K')liBi{t)+ {t/mJ)B,m,

711+W = [_m^/{mJ-^XK')TBi{t)+ {t/mJ)B^m,

710+(0 = ^ 4 ( 0 .

The Bi are now free from all kinematic singularities and zeros. In the absence of specific dynamical informa­tion, Bi can be assumed to be smooth functions of /.

For yp —> 7r+^, most of the above discussion can be carried over except now the two baryons have the same mass.^ Equations (1) and (4) remain valid if we replace K by x, Kc by TTC, W+ by 2mN, and set w _ = 0 . The important fact is that the two constraint equations (2) and (3) now occur at the same point {t—mJ-=0), and the two conditions collapse into one. We get

/ io -=w+/ i i+ , a t / = 0 (6)

while / i i~ completely decouples from the above two amplitudes. We have, therefore, only one constraint equation in the (baryon) equal-mass case. We para­metrize the residue functions for this case as follows:

7io-(/) = i 5 / - f 5 / ( l - / / / . , 2 ) ,

yu-it) = B/,

7lO+W=54'.

(7)

I t is easy to see that with this parametrization Eq. (6) is automatically satisfied and the B/ are free of all kinematic singularities and zeros (constraints). The first equation of (7) is written in that particular form so that at /=At/ the residue function reduces to Bi which is to be compared with the value obtained from elementary ir exchange.^

With the parametrization (5) and (6) we fit the most recent data from SLAC on high-energy forward photoproduction of 7r+ and K+ mesons. As our primary purpose is to study the implications of the kinematic constraints (2), (3), and (6) in the parity-doublet exchange model, we shall confine ourselves to the small-/ region where these constraints are important. We shall also limit the number of free parameters to a minimum in order to bring out the essential points. It is obvious that increasing the number of free parameters can only

7 Obviously, we neglect the small electromagnetic mass differ­ence between the proton and the neutron.

^ A similar consideration could be applied to the previous case of K production. However, in that case t=ix]^ is a far-away point as compared with the range of / values we consider. Therefore, we did not explicitly exhibit the value of 7io~ at t—^iK^.

0.0 -0.02 -0.04 -0.06 -0.08 t (GeV )̂

-0.10

FIG. 1. Fits to the forward yp -^ ir'^n scattering. The soHd curves are fits obtained with Q:' = 0 . 6 1 , ^ I / 5 = 0 .65 , B^/B^-0.21, B3=Bi — 0. The dashed curves are fits with parameter values of Ref. 2, a'^1.0,Bi/B = 1.0, B^/B^^-OA, where B = eg{ir/^)n^\

improve our fit. Thus, we take the energy-scaling factor .̂ 0 to be 1 GeV^ and set the kinematic-singularity-free and zero-free residue functions Bi and B/ to be con­stants. We also take a(t) = adt) == a^(t—/JL^) in both cases. Since the amplitude fio^ for K production and the amplitudes fif and fio^ for TT production do not enter into any of the constraint equations, they have constant residue functions in this model. On the other hand, the residue functions for the other amplitudes must have at least linear dependence on / due to the kinematic requirements. These /-dependent residue functions, having zeros at close-by points, are rapidly varying in the small-/ region. I t is these latter ampli­tudes that are important for the presence or absence of forward peaks in the differential cross sections. To reduce nonessential free parameters, we therefore set the residues for the constant amplitudes to be zero (le.,Bi=Bz' = B/ = 0),

Good fits are obtained for both processes at almost all energies ranging from 5 to 18 GeV. The fits are not particularly sensitive to any of the adjustable parameters chosen. The results for yp —> ir'^n are given in Fig. 1. Solid lines represent our best fit. Dashed lines represent fits obtained by using the parameters quoted in Ref. 2. The results for yp —> K^A are given in Fig. 2. In this case we have experimented with a variety of choices of parameters. We found that the general features of the data can be equally reproduced by a wide range of parameters as long as one does not set all 7's to be constants. Figure 2 represents one of such possible fits with Bs set equal to zero. These results

Page 3: Mesons in the Regge-Pole Model

1448 C . C . S H I H A N D W U - K I T U N G 180

r.o

J 5 GeV

8 GeV

II GeV

0.0 -0 .02 -0 .06 - 0 . 0 8 t (GeV^)

- 0 . 1

FIG. 2. Fits to the forward yp -> K^L scattering with Q;' = 0 . 6 1 , 5iVi5' = +0.11, ^ 2 ' / ^ ' = - 0 . 1 2 , ^ / = ^ / = 0, where

clearly indicate that the parity-doublet exchange model can explain both the TT- and iT-photoproduction data despite the apparent difference in the shape of the forward differential cross section.

Several additional remarks on our results are in order.

(i) Previous attempts to fit the i^-production data in this model failed because they assumed all 7's in (5) to be constants. The fact that the 7's must satisfy two constraint equations, (2) and (3), at very close-by points makes this assumption rather artificial. I t results in an oversimplification of the kinematics and thus failure to fit the data. We have shown that it is much more natural to assume the kinematic-singularity-and zero-free quantities Bi and Bi (or other sets equiv­alent to those chosen) to be smooth. This indeed leads to good agreement with data. Our success in attributing the difference in observed differential cross section for TT and K photoproduction to kinematics alone helps to remove a very unpleasant feature for the parity-doublet exchange model. This does not necessarily mean, of course, that we have established this model as providing the true mechanism for these photoproduc­tion processes. At the present stage, it does not seem possible to either prove or disprove the model in a de­finitive way.

(ii) As more data are now available for 7/? —» i^+S^, we have also tried to fit these data with the same model. Again reasonable fits can be obtained. However, here the error bars are still too big to allow one to draw definitive conclusions. Consequently, the detailed re­sults are not presented here.

(iii) One argument which has been used to support the inclusion of a /f * contribution is that experimental data indicate^ comparable forward cross sections for S" and A production in yp~^KY while the ratio of coupling constants, gKi^p/gKKp, seem to be small from dispersion-relation analysis of low-energy data.^ Besides the uncertainties that still beset the determination of the coupling constants, this argument rests on two assumptions: First, the observed forward differential cross sections are dominated by the /oi~ amplitudes; second, the residue functions 7oi'~(0 for both A and 2 production extrapolate smoothly from / = 0 to t~ixj^, Our fits show that /oi~ almost never dominates in the forward direction. The other amplitudes are at least comparable in magnitude to /oi~. More importantly, as we have emphasized many times, the kinematic con­straints demand that the 7's have rather strong / dependence near / = 0 . (Even if the 7's are smoothly varying for both cases, an increasing 701- for A and decreasing 710" for S away from /==0 can produce a rather large variation for their ratio as we extrapolate it toward i—ii^^ Since t=ixj^ is a rather far-away point (at least by our scale), it would not be surprising that the ratio of the residues 701" for 2 and A production could change significantly from / = 0 to t—ixj^. Therefore, we do not believe that there is an inconsistency involved. To put this in another way, from our experimentation with the possible range of fitting parameters, it was clear that we could accommodate a small gK^p/gK^p ratio (say 1/7) by imposing this as a constraint at t—jjLK^,

(iv) Because of our specific purpose, we deliberately limited our attention to very small values of t where the kinematic constraints are important. This enabled us to eliminate many inessential free parameters in the model, and bring out the important difference between the baryon equal- and unequal-mass cases. To fit data at larger values of / many more adjustable parameters have to be included. In order to make detailed fits it is best to consider not only one particular reaction but all other reactions which are related to it by factorization. In this connection, we mention a recent work toward this direction for pion photoproduction.^^ I t is worth point­ing out that even in our simplified model the nonvanish-ing parameters Bi and B/ are not uniquely determined. Thus, although the linear 7xju'̂ (0 in our model always imply zeros of these functions in the neighborhood of / = 0 , we cannot tell precisely in which amplitude (s) they occur or precisely where they occur. (Many different sets of parameters give comparable fits.) To pin down these zeros, simultaneous study of related processes like yp —> K^A, irp —> iDV, etc., would be very helpful.

We wish to thank Dr. A. M. Boyarski for sending us recent data from SLAC prior to their publication.

9 J. K. Kim, Phys. Rev. Letters 19, 1074 (1967). 10 R. C. Brower and J. W. Dash, Phys. Rev. 175, 2014 (1968).