175 Q U A ' S I D I F F R A C T I O N S C A T T E R I N G 2191

W gets large (compared with the masses). R(W) is the explicit contribution of the spins. Since ec^(W2—mi2)/ 2mi, the dependence, on W is as PF~4 for large W (if JR=1). The result is a peaking at some intermediate value. The peak may in principle be very sharp, and near threshold ("Deck effect"). For example, if mi=w2

==m (nucleon mass) and m3=M (pion mass), then e~2

oc (pp—0^-2^ while threshold is at W~nt-\"V>, so that there is a very sharp peak just above threshold. How­ever, this can be qualitatively modified by R(W), which for | + - » | + + 0 - behaves as (W2-m2)2/W2. Thus the net behavior in this case is as {p%w/W)W~2, which displays a broad enhancement at FF^d400 MeV

TO explain certain observed features of reactions in which pion exchange is allowed, the hypothesis

of the existence of a conspirator trajectory with positive parity and with other quantum numbers the same as those of the pion has been made.1 In particular, this has made possible an explanation of the forward peak in 7r+

photoproduction.2,3 Rough estimates of the pion and conspirator trajectory functions have been made by different authors, using finite-energy sum rules for photoproduction.4 Recently, alternative mechanisms5

have been proposed for explaining the observed features of pion photoproduction, and it is therefore desirable to make a further check of the conspiracy hypothesis.

We have recently used the continuous-moment sum

* Research supported in part by the U. S. Atomic Energy Com­mission (Report No. NYO-2262TA-190).

1 R . J. N. Phillips, Nucl. Phys. B2, 394 (1967); F. Arbab and J. Dash, Phys. Rev. 163, 1603 (1967).

2 A. M. Boyarski et al.t Phys. Rev. Letters 20, 300 (1968). 8 S. Dreli and J. Sullivan, Phys. Rev. Letters 19, 268 (1967);

S. C. Frautschi and L. Jones, Phys. Rev. 163, 1820 (1967); J. S. Ball, W. Frazer, and M. Jacob, Phys. Rev. Letters 20, 518 (1968).

4 A. Bietti, P. Di Veechia, F. Drago, and M. L. Paciello, Phys. Letters 26B, 457 (1968); D. P. Roy and S. Y. Chu, Phys. Rev. 171, 1762 (1968).

8 D. Amati, G. Cohen-Tannoudji, R. Jengo, and Ph. Salin, Phys. Letters 26B, 510 (1968); J. Frjzfyland and D. Gordon, MIT Cambridge Report (to be published).

and is oc W~~2 for large W. Higher spins would result in an J? (IF) involving larger powers of W, so that the peaking can not only be broadened and shifted, but even eliminated. The particular case will depend on the spins and masses involved.

ACKNOWLEDGMENTS

I am grateful to Dr. D. J. Brown and Dr. M. K. Sundaresan for some useful conversations. Part of this work was done while I was a guest at the University of Tel Aviv and I am grateful to Professor Y. Ne'eman for his kind hospitality.

rules6 for determining the A% trajectory and residue functions in pion photoproduction.7 As we have ob­served in our earlier work,7 these sum rules provide a more reliable method of determining the Regge tra­jectory parameters than the finite-energy sum rules; and in this paper we use them for examining the ques­tion of pion conspiracy in pion photoproduction. Here we use these sum rules, assuming pion conspiracy, to obtain estimates for the trajectory and residue functions of the pion and the conspirator. The nature of the results thus obtained enables us to examine the validity of the pion-conspiracy hypothesis. Further, we examine in detail how the results obtained from the continuous-moment sum rules depend on the value of the moment parameter y; this gives an estimate of the reliability of the results obtained using the available fits to the photoproduction amplitudes.8

8 Y . Liu and S. Okubo, Phys. Rev. Letters 19, 190 (1967); A. Delia Selva, L. Masperi, and R. Odorico, Nuovo Cimento 54A, 979 (1968).

7 K. V. Vasavada and K. Raman, Phys. Rev. Letters 21, 577 (1968).

8 We have used the fits to the multipole amplitudes obtained by R. L. Walker et at. (to be published).

P H Y S I C A L R E V I E W V O L U M E 1 7 5 , N U M B E R 5 25 N O V E M B E R 1968

Continuous-Moment Sum Rules and Pion Conspiracy in Photoproduction

K. RAMAN*

Physics Department; Brown University, Providence, Rhode Island 02912

AND

KASHYAP V. VASAVADA

Physics Department, The University of Connecticut, Storrs, Connecticut 06268 (Received 24 July 1968)

Continuous-moment sum rules are used for examining the validity of the pion-conspiracy hypothesis in pion photoproduction. The trajectory and residue functions of the pion and conspirator trajectories are estimated.

2192 K. R A M A N A N D K. V. V A S A V A D A 175

We define the amplitudes

<?<<-> (M) = ^y if- vo2)-^^ fat),

side of (3) vanishes. For »= 1, we obtain the sum rules

(i)

where f=2, 3, v^voiti^P+Q+ifi/Atm, ftH=0iH

+M2(~ ))? and F3

(-)=-»(2w^i<->-Mo(~)).9 In using the pion-conspiracy hypothesis, we assume the following high-energy behavior of the amplitudes F2(r~) and Fz^ for small £ (and large v):

Fi^foh+Piitfait} %(!+&-**"*)/vy*"1

simrai W ? (2)

which will hold if a single Regge trajectory dominates each amplitude F^ at large v, for small (fixed) L In Eq. (2), *=2, 3, A(*)=0,WG**f-fl-S ft(0=ft(4 ^W—^W? «S (0 = = «C(0 / ° &n(i ^i is a scale parameter, conveniently chosen to be 1 GeV.

Following well-known methods (see, for instance, Ref. 7), we obtain the following sum rules:

1 f K J PQ

dp (v2- pi)-y{ (coswy) ImF<<->

+ (sinry) ReFi<->)+ri(vi?-v9*)rr

simr(f a—7) cu(t)pi(t) V***

27rsin|7TQfi a»— 2y j ^ * " 1

From these we obtain

cfy 2T sin Gbra*) (a,-—-27) *>ia*wl

X[(ia»—r)"1—«" cotwQai- y)—2 In*].

We note that

(3)

(4)

= [ ( l a i - 7 ) " 1 - ^ cofor(Jar-7)--2 InP]"1; (5)

the right-hand-side of this equation depends on the trajectory function ai(t) but not on the residue func­tion 0i(t).

InEq. (3),

eg (iJ?+f) eg f2=s f 3 = — _

4m Qi2—t) 4/» C2f»+0*i»—Mn)fl. (6)

As in Ref. 7, we note that when, for given t, 7 = 7 ^ ^ ) —^m(t)—nf with # = 1 , 2 ,3, • • • then the right-band­

$ 7

l o<(/)ft(0 P2

4 sinf?ra,(0 (̂ i)a*~

(7)

(8)

9 For the notation and kinematics, see for instance j . S. Ball, Phys. Rev. 124, 2014 (1961).

10 We note here that F3 ( - ) receives a contribution from the A% trajectory also. The simple form (2) for the high-energy behavior of Fz^t keeping only the conspirator trajectory, will be a good approximation only if the A 2 residue function in Fz^ is negligibly small for small L

In principle, one could solve (3) for any pair of values of 7, for fixed t, and obtain a(t) and fi(t). In particular, 7 may be chosen so that the contributions of ReFV"^ to the integrals are much less important than the contri­butions of ImFi^K (This would minimize the errors arising from the real parts of the amplitudes, which are subject to much larger uncertainties than the imaginary parts.) For instance, in the finite-energy sum rules,11 one chooses values of 7 equal to 0 and 1, respectively.

However, choosing different values of 7 weights differently the various energy regions contributing to the integral. Because of the uncertainty in the available data for pion photoproduction, the choice of 7 can significantly affect the results, and it is not clear that one would obtain correct results by solving (3) for two values of 7 differing widely, as is done in using the finite-energy sum rules.

The most appropriate procedure appears to be to solve (3) and (4) simultaneously for the same value of 7, repeating this for various fixed values of 7, for each Ln [As a particular choice, one may take Eqs. (7) and (8); the value yA of 7 at which $(t; P,y) vanishes, for fixed I, then gives a(t) through the relation a(/) = 2yH-2.]

We remark that now the real part of the amplitude plays an important role in determining the results obtained from the sum rules; this is unavoidable if one is to examine how the content of the sum rules (3) depends on 7.13

We have solved (3) and (4), for values of 7 ranging from —2.5 to 0, to obtain aT(t), fiv(t), ac(t), and @e(t) over a range of ^between —10 #2 and +10/*2? taking P=zv(i) corresponding to an incident photon laboratory momentum of 1.2 GeV/c.14'15

In Figs. 1-4, we give the solutions for av{t)y &•($> ac(t)} and 0c(t) for different values of 7.18 We have not

11R. Dolen, D. Horn, and C. Schmid, Phys. Rev. Letters 19, 402 (1967); Phys. Rev. 166, 1768 (1968).

m This is done by solving Eq. (5) for at** (I) and using this in (3) to obtain fii(t).

18 We remark here that for a range of t such that Qo> (*)<- 1] is close to an integer, Eq. (7) enables a determination of a(t), using particular values of 7, such that the real part of the amplitude plays a relatively unimportant role. Correspondingly, however, in the determination of 0(0 from (8), the real part of the amplitude plays a more important role.

14 The amplitudes F*(~> were obtained using the multipole fits obtained by R. L. Walker et ah (see Ref. 8), which are given up to a value of the incident photon laboratory momentum equal to 1.2 GeV/c.

15 We have also independently solved Eqs. (7) and (8) and have checked that the results agree with those obtained from (3) and (4) for the particular values of y involved in (7) and (8).

16 These of course satisfy the kinematical constraints «c(0) =0,(0) and 0,(0) = - Q*„2/2m)(3e(0).

175 C O N T I N U O U S - M O M E N T SUM R U L E S IN P H O T O P R O D U C T I O N 2193

i -10

..>*•

-0 .2

a a j t )

o

i

-0.1

^&*^ -5 2J*^

°^f\ ^ ^ /

-^Z-^ / -̂* /. /

/

/ / / / / / / /

—0.1

- - 0 , 2

—0.3

(a)

^ s * ^ * \

5 r >°

o SOLUTION (a) v SOLUTION (c) A SOLUTION (d) a SOLUTION (e)

FIG. 1. The solutions for the trajectory function av{t) for 7=-2 .5 , —1.5, —1.0, and -0.5 [denoted by (a), (c), (d), and (e), respectively]. Solution (b), for 7= —2.0, is very close to solu­tion (a) and is not shown separately. * is in units of nv

2.

shown the solution for 7 = — 2.0, as it is found to be very close to that for 7 = —2.5. Further, for /3T(/), the solu­tions for 7 = — 2.0, —1.5, and —1.0 are very close to the solution for 7 = —2.5, and we have therefore not shown them as separate curves.

On the whole, the results obtained for 7 in the range — 1.0 to —2.5 are roughly consistent with one another. The results for 7=—0.5 differ considerably from the results for the other values of 7, particularly those for a„(t) for moderately large negative /.

For small negative /, most of the curves shown here are approximately linear and it is instructive to try a linear parametrization of the results for such values of /. An average of the results for 7 = — 1.0 to —2.5 gives

2 . 0 -

1

1.0-

-10 - 5 J I 1 ^^

— 0 — " " " ' 1 - ^ ^ ^ ^ ' ^

-1 .0 -

/ ( e )

// //

X 5 10

t

0.4_

0 .2 -

- • • ° - ?

.^jf/ -0.2-

^ ^ - 0 . 4 -

, a c < "

10

t

FIG. 3. The solutions for ae(t). (Notation as in Fig. 1.)

the following, for small negative /:

«7r(0«(-O.O3O±O.OO7)+(O.O21±O.OOl)^, (9a)

ftr(*)»( 0.133db0.006)+(0.108=1=0.005)/, (9b)

<*.(/)~ (-0.037±0.006)+ (0.035±0.007)/, (9c)

&(0«-.(1.78 =fc0.06 )-(0.020=1=0.008)/. (9d)

The pion residue function goes through zero at a value of / of about — l.lju2 to —1.2/*2. The conspirator trajectory passes through zero at /«0.85ju2 to l.l/*2; the conspirator residue function is not zero at this point. This indicates that the conspirator trajectory chooses "nonsense," in agreement with the conclusions of earlier workers (e.g., see Ref. 4).

To compare our results with those of earlier workers, we first note the results of Bietti et al.A:

«,(/)»-0.002+0.004/-; «,(/)«-0.002+0.006/;

ftr(0«0.15+0.1/. (10)

Ball, Frazer, and Jacob3 assume

ar(/)=a«(/)« -0.02+0.02/,

FIG. 2. The solutions for the pion residue function £*(/). (Nota­tion as in Fig. 1.) Solutions (b), (c), and (d) are very close to solution (a) and are not shown as separate curves. As most of the points in solution (d) are very close to those of solution (c), they are not shown separately.

1 1 -10 - 5

~"^:^3^

0

-

-1.5-

1. r-" r 5 * 10

k £c(t)

S N ' ^ : ^ C N ^ _ - 2 . 0 -

-2 .5-

FIG. 4. The solutions for &,(/). (Notation as in Fig. 1.)

2194 K. RAMAN AND K. V. VASAVADA 175

and obtain &T(t) as given in (10), (In all the above results, t is given in units of /*T

2.) Comparing these with our results, we see that our

values of aT(0) and ac(0) differ appreciably from those obtained by Bietti et al. Also, we obtain slopes for aT(t) and ac(t) that are considerably larger than those in the estimates of Bietti et al; the slope of f$v(t) as obtained by us is about the same. The slope of aT(t) as obtained by us is about the same as the "canonical" slope (of 1 GeV~2) assumed by Ball, Frazer, and Jacob, while the slope of ac(t) that we obtain is of the same order.

In order to examine the effect of choosing a larger value of P, we have extrapolated the results of Walker et al. to an incident photon laboratory momentum fc= 1.5 GeV/c and solved the sum rules, with the cor­responding values of v [ = v(t)~], to obtain aw(t)f flv(t), adt), and fic(t). We find that the results for the inter­cepts at £=0 are about the same (within about 10%) as the results obtained with fc=1.2 GeV/c. The slopes of ac(t) and pp(t) differ by about 30% on varying k from 1.2 to 1.5 GeV/c, while the slope of 0r(t) is altered considerably more. The solution for av(t) becomes much more uncertain when one takes fc=1.5 GeV/c. When fits to the multipole amplitudes become available up to higher energies, we may examine more reliably the effect of varying v in the sum rules (3).

The existence of consistent solutions for ae(t) and Pc(t) (for 7 lying between about —1.0 and — 2.5) shows that it is a good approximation to parametrize the high-energy behavior of F3

(~) in the form (2). If (2) is re­garded as the high-energy contribution of a single Regge trajectory—the conspirator trajectory—then our re­sults give the trajectory and residue functions of this trajectory, and suggest that the contribution of other trajectories, in particular the A% trajectory, to Fz^ is small in comparison. Our result for ac(t) gives a tra­jectory that is lower than the A^ trajectory (as given by the available estimates); this suggests that the residue function for the At contribution to F 3

( - ) is negligibly small for small values of t.

In examining our results as a function of the moment parameter 7, we have noted that the results obtained with different values of 7 are reasonably consistent with one another for values of 7 between about —1.0 and — 2.5. For 7 = —0.5, the solutions deviate considerably from those obtained with the more negative values of 7 ; this is even more so for 7=0 . This shows that it is important to take into account how the choice of 7 may affect the results; further, it throws doubt on the cor­rectness of the results obtained (in pion photoproduc-

tion) using 7 = 0 and 7 = 1 simultaneously, as done with the finite-energy sum rules.

We interpret the variation of our results with 7 as indicating that the uncertainty in the available multi-pole fits used here is large in some energy regions, and further that choosing 7 between about —1 and —2.5 gives a greater weight to the more significant and better-known energy regions in the dispersion integral (e.g., the regions where the resonances occur). The deviations that arise on choosing 7 more positive than about —0.5 may be due to an uncertainty in the multipole fits in the low-energy region.

To sum up, in this work we have used the continuous-moment sum rules to examine the validity of the pion-conspiracy hypothesis in pion photoproduction. Our results indicate that the parametrization of the ampli­tudes for large energies and small / required by the conspiracy hypothesis gives consistent solutions. This supports the validity of the conspiracy hypothesis; however, with the uncertainties in the data available at present, one cannot rule out more complicated mecha­nisms. The uncertainty in the available data introduces a corresponding uncertainty in our results for the func­tions aT(i), pT(t), ac(t)9 and &,(/); however, we believe that the qualitative features of these are given correctly by our results.

Further work on the use of continuous-moment sum rules in photoproduction will be reported elsewhere.

We are grateful to Professor R. L. Walker for sending us his preliminary results on the multipole fits. Part of the computation in this work was carried out in the computer center of the University of Connecticut, which is supported in part by Grant No. GP-1819 of the National Science Foundation, and part at the comput­ing laboratory at Brown University.

One of the authors (K.V.) wishes to acknowledge partial support of research by the University of Con­necticut Research Foundation and the award of a summer faculty fellowship. He also wishes to thank Professor K. T. Mahanthappa for his kind hospitality at the Summer Institute in Theoretical Physics at Boulder, Colo, where he was during the final stages of the work.

After this work was completed, we received a report by P. Di Vecchia et al. (International Centre for Theo­retical Physics, Trieste) dealing with the same prob­lem.17 Our work differs in detail from those of these authors.

17 P. di Vecchia et al., International Centre for Theoretical Physics, Trieste (to be published).