., SLAC-PUB-655 Deceber 1969 (TH) and (EXP) REGGE AALYSS OF ro AD n+ PHOTOPRODUCTO AT BACKARD AGLES J. V. Beaupret and E. A. Paschostt Stanford Linear Accelerator Center Stanford University, Stanford, California 9435 ABSTRACT + By correlating recent experiental results on 'T' and 'T photo- production at backward angles with results on 7r-p-p~- at the sae angles, it is found that any explanation of the three processes in ters of Regge poles ust involve at least two isospin 1/2 trajectories. Furtherore the two trajectories ust be alost degenerate. Such a solution akes definite predictions about other processes which can be checked by experients. (Subitted to Phys. Rev. ) ' ork supported by the U.S. Atoic Energy Coission A. E. C. Postdoctoral Fellow. Aachen Technischcn Hochschulc, Aachen, Gerany. "Present address: Rockefeller University, ew Yorli, ew York. d
TRODUCTO 1 + The first data on the photoproduction of r esons at backward angles iposed restrictions on the contribution of various trajectories. These restric- tions becoe ore liiting now that there are accurate data3-5 available for the + processes n, --p~, w-pn and n-p -pp-. n fact by correlating these proc- esses one arrives at the conclusion that an analysis of the data in ters of traditional Regge poles is non-trivial and it requires the contribution of at least three trajectories. Furtherore, the contribution of the -- 3/2 trajectories to photoproduction has an upper bound and the = 1/2 trajectories ust be alost degenerate. n view of the above results we feel it is justified to neglect for the oent the contribution of the absorption cuts, which has not yet been forinulated copletely, and deal with a three pole odel which leads to definite conclusions. These conclusions can easily be checked experientally. n case that they are violated we feel that a siple pole odel is inadequate and ipractical and that the contribution of cuts is essential. n the next section we give the basic Regge foralis and relate the residues 7 to the coupling constants of the Born diagras deterined in the isobar odel. n Section 111, we suarize the experiental situation that requires us to introduce the -trajectory, we obtain fro vector eson doinance an upper bound Y for the contribution of the = 3/2 trajectories to photoproduction and give the paraetrization of the aplitudes. The last section discusses the ain results of the odel and proposes experiental checks. -2-
. FORMALSM The kineatics of photoproduction have been discussed in several places. 8 e adopt the C. G. L.. notation. The C. G. L.. aplitudes have kineatic singularities, which becoe apparent when one writes the in ters of the 9 J. Ball aplitudes which satisfy the Mandelsta representation: (11. lb) (. C) (. d) where =&, k = (u-m3/2 El f M = (lq2/2, E2 f M = [(amf - p2]/2 The cosine of the u channel scattering angle is give11 by: U e =- U ~ S U 2 2 2 2-2M U+ (u-m )(U+M - p ) -3-
The phase conventions for the several square roots appearing above are these: All expressions of the for (-af12 have a branch point at =a and the cut lies along the negative real (-a) axis. Slightly above the cut the phase of (-af12 is 7r/2. Products like (2-a2)l12 have cuts between +a and -a, and they satisfy: Thus in the backward direction (zs=-1) one has zu=-1, and at u=o, zu=+ 1. The fact that zu is not large for backward angles is not very alaring since a sequence of daughter trajectories can restore the Regge asyptotic behavior. A. McDowell Syetry Since the Bi(s,u) are even functions of, we can derive relations between gi() and Si(-). ith the phase convention adopted in the previous section we obtain: s,(w, = q-w, (.3a) (. 3b) B. Singularities at u=o Since the line u=o is within the physical region, special care ust be taken so that the cobinations of sines and cosines of OU/2 together with the linear cobinations of the si() which appear in the helicity aplitudes are sooth around the line u=o. Assuing that the Bi aplitudes approach constants as -, we find that the ost singular behavior of the sus and differences is given by qw) + F2()-C1/ 2-4-
. (. 4b) (n. 4 ~ ) (U The trigonoetric functions of 8,/2 behave as cos Ou/2-1. Therefore the following u-channel helicity aplitudes vary soothly over the line u=o; 1 = -Ji nw (g3 +g sin eu COS - u A1/2, 3/2 4 2 u (n. Sa) All2, A-1/2,3/2 = h T (2 p2 - S1] = fi nw ( g3 - s4) 1 cos 5 Bu + sin u U 1 sin 3 eu (. 5c) C. oralization The differential cross section is given by where ks is the center-of-ass oentu in the s channel, and A w the u channel aplitudes, ~ (p) being the final (initial) center-of-ass helicity. At 18, where zs = zu=-l, only the A-(1/2, 1/2) aplitudes is non-vanishing. n case that the contribution of the other aplitudes is non-negligible we expect a iniu at 18. -5-
D. Reggeization The Reggeization of the aplitudes incorporates all of thc above properties. Because of the McDowe11 syetry, we Reggeize only c92 and. F4; g1 and S3 are obtained by reflection in. n ters of the electric and agnetic ultipoles, 1 M () and E (), with j = 1 f -, the partial wave expansions are given by: l l 2 (11.7a) The cobinations lm1,!el-, (1+ l)e1+, (f+ l)m can have dynaic poles in the - 1+ coplex j plane. For a given Regge pole at j = a, the leading contributions coe fro M 1- and El-. Thus we keep only the lm1- in S2. Assuing (11.8) we obtain the Reggeization: (11.9) For 9 we assue 4 (11.1) and obtain: (11.11) -6-
ntroducing the scaling paraeter so and extracting the kineatic singularities, we obtain a set of residues yi() which contain only dynaic inforation: (. 12a) (11.12b) The resulting for of the Reggeized aplitudes are E. Born Diagras e evaluate the residues of the three leading trajectories at the poles by using Born diagras. The nucleon residue at the pole is obtained by using Feynan rules and then using Eqs. (11.8) and (.12a) to obtain (11.14a) (. 14b) where 2 G /4n= 14.5, e 2 /4n = 1/137, p = 1.78 e/2m, and p = -1.91 e/2m. P n n estiating the residues of the y and the A trajectories we use the isobar 7 odel of Gourdin and Salin. The contribution of the y resonance to tho M () 2- -7-
(11.15) where L2 =.l(?)(a) 4n 4n JG- (11.16) n (11.15) we have corrected for the noralization differences between references 7 and 8, but we have not included any isotopic spin factors. Using (11.15), (11.8) and (11.12a) we can obtain y2($. n a siilar way we can estiate the A-residue. 7 The nuerical results using the coupling constants of Gourdin and Salin are:. EXPERMETAL SUMMARY AD PARAMETERZATO The ' n and no photoproduction can be cobined in such a way that one can ake rather strong arguents about the contributions of the allowed baryon exchanges. On the other hand, application of vector eson doinance iposes additional constraints on the contributions of the trajectories. There are three rather iportant features of the data and we discuss the in detail: 1. 2 The traditional nucleon trajectory has a nonsense zero at u = -. 15 (BeV/c), which appears in all four helicity aplitudes. The absence of a dip in this u region in both thc no and n+ photoproduction eliinates the doinance of ihe u! trajectory. Presuably this arguent is weakened if the nucleon contribution is coposed of 6 the usual Regge contribution and an -Poeron cut; or if therc is n fiscd polc in CY the residues at a( ) = -1/2. a! a! Since the undcrstanding of cuts :tiid fiscd poles is rather liitcd, we restrict our discussion to conventional Reggc pol~s. -8-
2. f one akes a conventional Regge odel containing only the a and the A poles, only the A contribution survives at the position the nucleon zero. Since the pure A(=3/2) contribution involves only the isovector part of the photon, the vanishing of the 1=1/2 contribution(s) iply (n. 1) The recent experiental data show that these two cross sections are equal for -. 3 < u -C.. Therefore there ust exist a non-vanishing =1/2 contribution at u 2 -. 15. Since we are forced to consider another =1/2 trajectory, we ay ask whether the A(=3/2) contribution is copletely negligible. The present data are consistent with an =1;/2 contribution alone, since one can vary the isoscalar and isovector ratios as a function of u to account for the observed cross sections. 3. The above arguent does not, - however, eliinate the =3/2 exchange contributions. Assuing vector eson doinance, there is an experiental way to estiate the =3/2 contribution to g~ -nn and yp--pno. This ethod relates the photoproduction contributions to the pure A exchange process n-p - pp-, by + d - dc [. - contribution to (n p-np';) = 2/9 (n p -pp-) (n. 2a) d 2 P, 2 [A contribution to du (w - nn'g = 1 (-) Pll(u) 4 2 Pp?P du - contribution to du (n p-npo)l (111.2b) Accurate knowledge of the helicity density atrix eleent pll(u) and of the 'y-p 5 coupling constant together with the n-p -pp- data can deterinc accurately the A contribution alonc to photoproduction cross sections. The knowledge of pll(u) is rather liited, but onc c still obtain an upper bound on the A contribution. -9-
+ Table 1 shows the upper bounds for n and 17" photoproduction obtained for P1(u) = 1/2, y;/4n = 1/2. Coparison of the upper bounds and thc experiental photoproduction cross sections show that the =3/2 trajectories could be a signifi- cant part of the photoproduction cross sections. n particular, A exchange could 1 1 account for ost of the n photoproduction cross section and about 4 2 of the + 17 cross sections. n suary, the arguents 1 to 3 iply that an =1/2 contribution, different fro the traditional nucleon, is needed. Arguent 3 does not iply that the A is necessarily negligible as was ephasized by G. Kane. 1 e therefore use not only the and A trajectories but also thc y trajcctory a in our paraetrization of the aplitudes. The y trajectory is thc necded =1/2 exchange suggested by arguents 1 to 3 and by the isobar odel, previously discussed. The trajectories are obtained as follows: The and A trajectories are taken fro previous paractrizations a of Refs. 11 and 12, to the reactions 7~ p-pn', n-p--pn- and r-p-pp-. The nucleon trajectory that we use is: + 2 a = -.34+ O.O93+ 1.15 (111.3) taken fro the work of Chiu and Stack. l1 For the A trajectory, we added a ter 12 linear in to Shih's deterination, so that the trajectory passes t.hrough the (1238) ass, to obtain: 2 a =.5 +.25 +.75 (. 4) The y trajectory is constrained to pass through the first two resonances ($152) 3/2-, 219) 7/2-) and also contains a ter linear in. The trajectory intercept was chosen as a free paraeter: a = x - (.385 + 1.15 x) + (.88 +.278 x) 2 (111.5) - 1 -
The residues $() have the following for: r4 = hu 1 c a (l+da)e (n. 6b) where a is an isospin factor which depends on the coupling of the various trajec- tories to the u channel vertices, and is given in Table 2. The paraeter S/V (see Table 2) occurs for the nucleon trajectories because the nucleons can couple to both the isovector and isoscalar coponents of the photon. e take this ratio to be a constant and the sae for all residues and =1/2 trajectories. As can be seen in Figs. 1 and 2, we obtained an excellent fit to the photo- production data. The chi-square was found to be 92.5 for 6 degrees of freedo. The quality of the fit did not change appreciably when a different nucleon trajectory, 13 2 1 = -.38 +.91 was substituted for the one given in Eq. (EL. 3). The rest of the paraeters are given by: a = -1.41 x a 1-3/BeV 2, : c = 1.1 x lo-2/bev 3, a = -3.66 x 1-3/BeV 2 A:, c = 5.25 x 1-2/BeV 3, a = 6.45 x y: 1-2/BeV 2, c = -8.2 x 1-2/J3eV 3, b = 8.43/BeV, d = -22.9/BeV. b = 7.78/E3eV, d = 1.97/BeV. b = O.O31/BeV, d = -.778/&V. 2 s =3.83BeV S/V= -.376 2 h = 4.19/BeV x = CY(~, ) = -.546-11 -
giving the y trajectory: 2 CY = -.546 +.22 +.73. V. COCLUSOS AD EXPERMETAL CHECKS An interpretation of the data in ters of three trajectories deands a delicate balance aong the. n fact, this analysis relies heavily on the degeneracy be- tween the CY and y trajectories, as well as the doinance of the A trajectory in soe regions. e suarize below those conclusions of the odel that can be checked experientally. A. -ydegeneracy -CY n the region -12 u 2<., the CY and y trajectories are alost degenerate. Such a degeneracy has already been pcoposed in the literature: 14 1. Several theoretical odels predict an a -y exchange degeneracy. 2. Both the absence of resonances in the pp syste and the absence of the CY dip in pp - D A+ can be explained in ters of degenerate a and y trajectories. l5 This degeneracy can be checked at places where the A contribution is sall by coparing w-nr + to the crossing syetric process En -7r- y. Furtherore a hotter check of the exchange degeneracy can be obtained by coparing the crossing + 15 syetric processes pp ~ D and T n-p-id. B. The A Contribution e find that the delta contribution is large and in soe places doinant, The percentages of the 1=1/2 and delta contributions to A production :kt pliotoii energies of 4 and 2 BeV are shown in Fig. 3. The upper bounds of the delta 2 contribution, given in Table 1, are exceeded by our odel for lul 3 13eV, - 12-3.
for the n+ data, and ost of the TO data. n the large u region where the y has a wrong signature iniu the delta trajectory is doinant. t would be very interesting to extend the n-p - pp- data to this region to deterine whether or not the delta contribution of the odel drastically violates the upper bounds. A large violation for a siple Regge odel would iply that the role of cuts is very iportant and can not be ignored in photoproduction. Another otivation for extending the T-p-pp- data to larger values of u coes fro the following observation. hen we separate the isovector contribution of the photon, by setting S/V equal to zero in our solution, we can relate it to p1 dcr/du(lr-p-np ) using the vector eson doinance odel. Such a coparison has been done by Z. G. T. Guiragossidn" who concludes that the agreeent is satisfactory at ELAB = 4 BeV and -1. < u <., but it is not very satisfactory in the larger u region. C. Photoproduction of n- at Backward Angles The y~-pn- cross section can be predicted within this odel. Figure 4 gives the calculated cross section at 8, 12 and 16 BeV. The p-pn- cross section is about two to three ties the - + nn cross section. This enhanceent is related to the S/V paraeter through the equation: dcr -(n du f ) = l2 (1 + S/Vf + T (1 + S/V) + lal 2, - 13 -
2 2 where (, T, lal are the contributions to the cross sections fro the nuclcon, nucleon-dclta interference and A respectively. Since S/V = - 376 fro our solu- tion and since the T is positive as it follows fro Fig. 3, we expect the 1T- cross section to be larger. At places where the ytrajectory is doinant the prediction that is rather genera1l3 and it should be checked experientally. D, Our solutions at 18' extrapolated to low energies pass through the ean of the differential cross sections. - 14 -
REFERECES 1. 2. R. Anderson et al., Phys. Rev. Letters 2l, 479 (1968). E. A. Paschos, Phys. Rev. Letters - 21, 1855 (1968). The proposal of this paper is the siplest choice in the absence of the nucleon dip. The new data on r-p-pp- do not allow such a large = 3/2 contribution. e take this opportunity to correct two isprints which occurred in the final printing of this paper: (i) n Eq. (1) the r-function f(a-1/2) should be replaced 3. 4. 5. 6. 7. 8. 9. 1. 11. 12. 13. 14. by f(a+3/2). (ii) n Eq. (ll), A should read like (.5b) in this text. 1/2,1/2 D. Topkins - et -al ' Phys. Rev. Letters - 23, 725 (1969). R. L. Anderson et al., Phys. Rev. Letters - 23, 721 (1969). E.. Anderson -2' et a1 Phys. Rev. Letters - 22, 12 (1969). J. V. Beaupre, Report o. SLAC-17, UC-34(TH), Stanford Linear Accelerator Center, unpublished. M. Gourdin and Ph. Salin, uovo Ciento - 27, 193 (1963); Ph. Salin, uovo Ciento - 28, 1294 (1963). G.?. Chew, M. L. Goldberger, F. E. Low and Y. abu, Phys. Rev. -' 16 1345 (1957). J. S. Ball, Phys. Rev. - 124, 214 (1961). G. Kane, talk presented at the inforal eeting on Processes at Backward Angles at SLAC (January 1969). - C. Be Chiu and J. D. Stack, Phys, Rev. 153, 1575 (1967). C. C. Shih, Phys. Rev. Letters 22, 15 (1969). V. Bargcr and P. eiler, preprint, University of isconsin (1969). J. E. Mandula, J. eyers and G. Zweig, Ply. Rev. Letters 23, 266 (1969); V. Bargcr and C. Michael, Phys. Rev. Letters 22, 133 (1969). - -
15. E. A. Paschos, Report o. SLAC-PUB-636, Stanford Linear Accelerator Center (August 1969). 16. Z. G. T. Guiragossit'h, Report o. SLAC-PUB-657, Stanford Linear Accelerator Center (1969).
TABLE CAPTOS 1. Coparisons of upper bounds for =3/2 exchanges with the data and calculated cross sections. The second and third coluns give the upper bounds for the contribution of =3/2 exchanges to the photoproduction cross sections. + Coluns four and five give the interpolated 'T and 'T experiental cross sections. Coluns six and seven give the percent contribution of the A to the calculated cross section. 2. The isotopic spin dependence of the aplitudes. The paraeter S/V is the isoscalar to isovector coupling ratio for the =1/2 exchanges, and is deterined by the fit to the data.
+; 3 5 8 $ 3 a 3 4 ( w. EJ dc " 4 b, -t n d " " " -t n. d. ". Y dc c- 4 E- O " 8 EJ ( d w ei dc. b,. EJ. -t dc c- Q, c-. 3 dc v) -t Q, 8 cll dc " d: d co. ( ell cll dc d b. E- d. dc Y b, " EJ hl. Q) co d 1. n d ". Q). C- Ln a, a, k 5 w B 5 a,.r( 8 c, a, G k 3 t-c c
Reaction TABLE 2 a A + yp- n7r 31, -Pro - h(l+s/v) (l-s/v) - h( l-s/v) 1 $z 1
FGURE CAPTOS + 1. The fit to T photoproduction. The photon oentu varies sornewkat about the indicated average. The theoretical points are calcu1:itcd using the photon oentu of each point. The data are fro Ref. 3. 2. 3. The fit to no photoproduction data of Ref. 4, The percentage contributions of =1/2 (,, + y) and =3/2 (A) exchanges + to the calculated.).r,-nn.rr cross section, The y signature iniu is 2 quite evident near u = -1.2 BeV. 4. The predicted cross section for.yn--pn- at incident photon cnorgics of 8, 12, and 16 BeV.
(u S > > Q ) Q ) o Lho; w w 4-+ % 1 l j 1 ll 1
u3-3 b w + 4 + - a 1 3 - l 1 1,.,. 1 1 1 1 L a
/ 1 / / / \ \ \ \ \ \ \ \
n \ - Y J a b vu cv @ - -