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Dips and spikes in dσdt for high energy exchange reactions
ANNALS
OF PHYSICS:
69, l-18 (1972)
Dips and Spikes in du/dt for High Energy Exchange Reactions* ARNON DAR Physics
Department,
Technion-Israel
Institute
of Technology,
Haifa,
Israel
Received August 3 1, 1970
A simple interpretation is offered for the origin of dips and spikes in do/dt for highenergy meson exchange reactions. Quantitative predictions are in good agreement with experimental measurements.
The differential cross section for two-body and quasi two-body final states in high-energy two-body collisions often exhibits a significant structure as function oft. In particular much attention has been focused on the dip at -t - 0.6 GeV2/c2 that was found in some reactions dominated by the even-signature vector meson exchange and on the dramatic narrow structure that was found in the small -t region, --t 22 mn2, of reactions dominated by w exchange. Naturally, many attempts have been made to understand these structures in terms of Regge Pole Exchange (RPE) phenomenology [l]: The dip at -t - 0.6 GeV2/c2 has normally been associated in the RPE model with the vanishing of the residues of the oddsignature vector meson trajectories at the nonsense wrong-signature point corresponding to the above t value. However, the seemingly erratic presence or absence of this dip (see Table I) has cast doubt on this explanation. Moreover, the simple Regge Pole Exchange Model with factorization also failed completely to explain the observed narrow forward structure in the cross sections for reactions dominated by 7~exchange. Some reactions which exhibit this narrow structure are listed in Table II. In this paper I would like to discuss a simple interpretation of the observed t dependence of cross sections for high-energy exchange reactions. A preliminary report on this work was included in a previous publication [23]. Here I will attempt to reformulate our basic assumptions in terms of general principles *Invited talk presented at the V-th “Recontre De Moriond,” Meribel-Les-Allues, France, March 9-20, 1970. Submitted for the Amos de4halit Memorial Volumes of Annals of Physics, Volumes 63 and 66, 1971, this article was inadvertently omitted from these dedicatory issues. Work supported in part by Stiftung Volkswagenwerk.
1 0 1972 by Academic Press, Inc.
595/69/r-1
2
DAR TABLE Reaction n-p a+n r-p a”P +P m+p +p
--f + +
7&l wp wn POP P*P WA++ +‘A++
YP - “““P yn + ?I% YP - TOP dp - K+Z+ K-p + Ron K+p - K”A+f
Dominant
exchange Dip at -i - 0.6 GeV2/c2 ? Reference
n-p --t p% 7T+n + pop n-P + Pir” a +P - P*P r+p -+ pod++ ~+p ---f pfrA++ dp dfoA++ a+n +fOp yp + n+n yn -+ r-p yp -n-A++ yn + +AK*p + K**p K-p --, K*n Kfp + K*A
Yes No No Yes Yes No Yes Yes Yes No Yes No No
2 3 4 5 6 7 7 8 9 10 11 12 13
Structure at -f ’ < m,,=
Reference
P P P 0 (TbJ P P w w
K”&O) P + P +
A 4
TABLE Reaction
I
Dominant
exchange
II
Dip Dip Spike Dip Spike Dip Spike Dip Spike Spike Dip Dip Dip Dip Spike
14 15 16 6 6,7 6 6 17 18 19 20 20 21 21 22
a ptr denotes p transversely polarized in the helicity frame.
rather than in terms of a specific peripheral model. I will then discuss a few representative reactions in some detail. A detailed comparison between theoretical predictions and a representative set of experimental results will be given. Finally, some comments regarding other approaches will be made.
3
HIGH ENERGY EXCHANGE REACTIONS
BASIC ASSUMPTIONS Our model is based on two main assumptions: (i) Exchange Reactions are “Surface Reactions” [24]; The S-channel partial waves which give the dominant contribution to an exchange reaction are the peripheral partial waves corresponding to s-channel resonances lying on an effective Regge trajectory j + l/2 - kR, where R is the hadronic radius and it increases only logarithmically with s, and k is the c.m. momentum in the S channel. This follows from Regge behaviour + duality [25]. This picture is also consistent with that underlying the absorption models, where the contribution of partial waves with j + l/2 < kR is strongly suppressed due to competition of many open channels. (ii) The peripheral partial waves of an exchange amplitude are approximately described by the corresponding partial waves of the pole closest to the physical region (the corresponding partial waves of the Born approximation for an exchange of the lightest particle lying on the exchanged Regge trajectory). This follows from the assumed analytical properties of exchange amplitudes together with the dispersion theory.
THE ORIGIN OF DIPS AND SPIKESIN du/dt FOR EXCHANGE
REACTIONS
What are the implications of (i) and (ii) for high-energy exchange reactions ? To answer this question let us examine the impact parameter expansion of an S-channel helicity amplitude for a reaction a + b ---f c + d with the helicity situation [h]; [A] = [A,, Aa, A, , A,]. It is given by FM = k2 fm b db J,,(b +t’)h(b). 0
h,+](b) is the contribution at impact parameter b to the scattering amplitude FI,] . J&b l/-t’), the cylindrical Bessel function of order dh, is a small angle and large j approximation for the rotation functions dL,(cosO), j being related to the impact parameter b through the classical relation j + l/2 - kb. p = A, - Xb,
and AX= t.~ - v is the total helicity change in the reaction. = t - tmin , where tmin is the minimum momentum transfer allowed by the kinematics of the reaction. According to (i),&(b) is appreciable only for b 2 R, i.e., v=X,-Ad t'
h](b)
s”;; 0.
(2)
4
DAR
According to (ii),
where Bl,] is the Born-approximation expression for the contribution of impact parameter b to the s-channel helicity amplitude with the helicity situation [h]. &l(b) for the exchange of a particle with mass m, and spin J, is given by
where (pe2 = m, 2 - tmin). Kdh is the cylindrical Bessel function of second kind and of order dA. It approaches rather quickly its asymptotic behaviour K&) &I
in the high-energy limit
1/( 7f/2x) e-=.
g&
(5)
depends only on the helicity situation,
but not on
b or on s.
When expressions (2), (3) and (4) are inserted in expression (1) they yield [26] hl
-
crAlw&JJ”-l
-
C~Al(s/s,-,)J6 dm
k
2
L,
b db JAb d-0
Kdb.4, (6)
e-“eR(“‘J,,(R
d/-f’)/(p,2
-
t’).
Consequently the t dependence of daldt is given by (7)
where [h] runs over all possible helicity situations.
NARROW
FORWARD
STRUCTURE
[27]
Let us first examine the behaviour of du/dt which results from TT exchange, in the forward direction. For small values of argument the cylindrical Bessel functions can be approximated by
Jk) so that da dt-
lxWA
XZA--zrY
dh positive
(8) (9)
HIGH
ENERGY
EXCHANGE
REACTIONS
5
If P,,~, the distance from the physical t region to the pion pole, is small enough, then the terms with AA = 0 on the r.h.s. of (7) when multiplied by the pion propagator will have a sharp forwardpeak, while the terms with AX # 0 when multiplied by the pion propagator will have a narrow forward dip. For high enough energy tmin -+ 0, so that P,,~ - mli2, and the type of the forward structure (a spike or a narrow forward dip, respectively) in du/dt depends only on the relative magnitudes of the different helicity amplitudes contributing to (9); i.e., on the relative magnitudes of the coefficients Cth] . These coefficients can be determined in the following way: Let us write the Born approximation for pion exchange in the form
where the Y’s are the vertex function for the corresponding It is then easy to show that the coefficients CL,,] are given by
Feynman diagram.
Table III lists the vertex functions which enter the reactions that are listed in Table II. In Table III eP is the spin-one polarization vector, e,,” is the spin-two polarization tensor, and u and U, are the spinors for the spin-112 and 312 particles. TABLE
III
cm vertex
1-o-o2+0-0-
2goazeii*@‘V p’ 4&.7? * m, e&f, QppppY
1/2+1/2+0-
&?w,
3/2+1/2+0-
$f
OYdP,
qp’h’)p’u(p,
4
#I)
We used the conventional definitions for the coupling constants, and the symbols p, p’ for the intial and final c.m. momentum, respectively. Explicit evaluation of the VA,,+in the high-energy limit p z p’ + co leads to the following rules:
(4 If I mc- maI > m, then I VA,&, m,“)l > I 6a,A,+, m,2)l for any h,h,h,‘h,‘, (b)
satisfying 1A,’ - A,’ j > 1A, - A, 1, If m, = m, and Jcp = J,’ = l/2+, then V,,, 1,2(~, m,,3 z 0.
6
DAR
These two rules are sufficient in order to correctly predict the observed forward structures that are summarized in Table II: (1)
r-p -+ pan, r+n + pop, r*p + pip, Kfp + K**p, K-p + K*n, rr+n-+fOp.
According to rule (a) the meson vertex prefers to conserve helicity, while according to rule (b) the nucleon must flip its helicity. Consequently the net helicity change is f 1, which leads to a narrow forward dip in daldt. (2)
n-p -+ p&n, yp + rr+n, yn + r-p.
There is one unit of helicity change at both the p$mr and the YTMTvertices. However, this helicity change is compensated for by helicity flip at the nucleon vertex [rule (b)]. Consequently the net helicity change is 0, which leads to a forward spike in duldt. (3)
r+p + pOA++, Kfp + K*A, rr+p + f OA++.
According to rule (a) both vertices perfer to conserve helicity. The net helicity change is therefore 0, which leads to a forward spike in du/dt. (4)
rr+p + p&A++, yp --f a-A++, yn + r+A-.
There is one unit of helicity change at both the p&mr and ynrr vertices while the baryon vertex prefers to conserve helicity [rule (a)]. The net helicity change is f 1, which leads to a narrow forward dip in dcrldt. Since experiments are performed at finite energies, the effect of tmin + 0 on the various cross sections and polarizations is worth studying. The most dramatic effect of tmin comes through pa2 in the pion propagator on the r.h.s. of (9), CL,,* being the distance of the pion pole from the physical t region. From the condition for a narrow forward structure in’daldt with a width of the order ~rn,~,
one can easily work out through the kinematics of any particular reaction the energy where one may expect narrow structure to start showing up. In particular, for reactions where mass change takes place only at one vertex, tmin --f 0 like @, while for reactions where mass change takes place at both vertices tmin -+ 0 only like s-l. We therefore expect the narrow forward structure in reactions of the first type to appear at relatively lower energies than those for reactions of second type. At lower energies t.~,~ increases, the forward structure becomes broader, and the forward cross section tends to decrease. For tmin > mlr2 no significant narrow structure is expected.
7
HIGH ENERGY EXCHANGE REACTIONS DIPS AND PEAKS IN THE INTERMEDIATE
--t
REGION
Let us now examine the intermediate --C region. The Bessel functions which appear on the r.h.s. of (7) can be recognised as the diffraction amplitude for radiation originating from a ring of radius R and with an angular momentum change Ah. These Bessel functions have simple properties as function of their argument and their order. Beyond their first maximum (around R d/--tl N Al) the cylindrical Bessel functions approach rather quickly their asymptotic form JLdx) xyA:dhmcos{x
- (Tr/2)[dh + l/2]}.
(12)
Consequently
[J&912 + [JddX)12 &
w7x)-
Since dc/dt is made of an incoherent sum of s-channel helicity amplitudes expression (7)] we may conclude from (7), (12) and (13) that
(13) [see
(a) Individual terms of the r.h.s. of (7) exhibit a diffraction pattern of maxima and minima corresponding to the extrema and zeroes, respectively, of expression (12).
(b) If a single helicity situation dominates the r.h.s. of (7), dujdt will exhibit the corresponding diffraction pattern. (c) The positions of the diffraction minima and maxima do not depend on the specific final state. They depend only on the parity of the total helicity change in the dominant amplitude on the r.h.s. of (7) [see expression (12)]. (d) The diffraction extrema are equally spaced when plotted as function of m [see expression (12)]. (e) The diffraction pattern shrinks logarithmically with increasing s, since R increases logarithmically as function of s. (f) If the helicity amplitudes with AX even and with Ah odd, respectively, contribute about equally to the r.h.s. of (7) we expect no “dip” structure in dojdt [this follows from (13)]. (g) Polarized cross sections should exhibit more structure than unpolarized cross sections. Rules (a)-(g) are the key to the “dip structure” of do/dt. We would like to stress however that all these predictions are only approximate. In general, both finite width effects and the imaginary part of the amplitude tend to wash out the diffraction pattern. In particular we do not expect any significant structure in da/dt at large -t values for reactions dominated by n exchange.
8
DAR
(Due to the small pion mass these reactions are poorly localized at a well-defined radius.) Note also that prediction (d) is in conjlict with simple RPE theory, since in the RPE models dips in du/dt are equally spaced as function of -t. We have seen by now that the presence or absence of a “dip structure” in do/dt for a particular exchange reaction depends on the relative magnitudes of all helicity amplitudes contributing to that reaction, i.e., on the relative magnitudes of the coefficients CL*] for all possible helicity situations. In the general case no simple rules can be given to determine the relative magnitudes of the coefficients CL~I , similar to the rules for pseudoscalar exchange. A general prescription for calculating these coefficients was given in Ref. [28] and [26]. The calculation in general requires (a) Knowledge of all relevant coupling constants. (b) Laborious Feynman algebra (Calculation of the Born approximation for one-particle exchange Feynman diagrams in the helicity representations and for all possible helicity situations). (c) Partial amplitudes.
wave expansion
of the s-channel Born-approximation
helicity
Many calculations of that type were reported in the literature concerned with absorption models. Below, we will analyze only a selected set of reactions for which analytic calculations are relatively simple, and for which the relevant coupling constants are known either from experimental decay rates or from reasonably successful symmetry schemes. (Generally in order to avoid tedious algebra and human mistakes, we recommend the use of the computer program that was described in Ref. [26].) 1. rr-p -+ rr% The reaction is dominated by p exchange. In the high-energy limit the ratio of the helicity flip contribution to the nonhelicity flip contribution to the r.h.s. of (7) is given by (m,2/4m,2)(GT/GV)2. From p-photon analogy [29] the ratio of the magnetic to electric coupling of the p to the nucleon is given by GTIGV = p9 - p,, - 1 = 3.7,
where pa and CL,,are the magnetic moments of p and n, respectively, in nuclear magnetons. Consequently, the term with dh = 1 in (7) dominates da/dt. For R N 0.8 fm it gives rise to diffraction dips at -t E 0,0.6,2.8, *** GeV2/c2, corresponding to the zeroes of J1(R V=?).
HIGH
2.
ENERGY
EXCHANGE
9
REACTIONS
Type I. rr+n -+ wp, r-p + wn, yp -+ q”p, Type II: rp -+ pp, yp -+ crop, yn - ran
The reactions in Type I are dominated by p exchange (note that g,,, > gw,,). The reactions in Type II are dominated by w exchange (note that gU,,” > g,,,,). For both types of reactions the helicity change in the meson vertex is f 1, because X, = f 1 and because a vector exchange couples only to helicity & 1 states of the external vector meson at the vertex. Ctnl for the four independent nonvanishing helicity amplitudes in the right-energy limit, are given by Cc,
1 l/21
1/2
C[o-
1/2
Go
112 I-
Go-
112
-
G”
+
1 1/21
-
GT
1/21
-
GT
-
GV +
I -l/z1
GT
G’
(Ah (Ah (Ah (Ah
= = = =
l), 0), 2),
1).
From p-photon analogy we know that GOT> G, v. Consequently for all reactions of Type I all CInl are about equal. da/& for these reactions should then exhibit no “dip structure”. From w-photon analogy we know further that G,” > GUT. Consequently for all reactions of Type II we expect the terms with 1Ah 1 = 1 in (7) to dominate du/dt. These terms give rise to the diffraction dips at -t’ = 0,0.6,2.8 .*. GeV2/c2 in analogy with r-p + non. Note that due to the terms with I Ah I = 2 the dip at -t’ = 0 is expected to be more pronounced, while the dip at -t’ - 0.6 GeV2/c2 is expected to be less pronounced than the corresponding dips in rr-p + non. 3.
r+p + rod++
The reaction is dominated by p exchange. p-photon analogy suggests that p is coupled to the pA vertex through a pure magnetic dipole coupling (StodolskySakurai Model [30]). This coupling favours unit helicity change at the vertex. Thus the terms with / Ah I = 1 on the r.h.s. of (7) will dominate du/dt. du/dt will then exhibit “dip structure” similar to the “dip structure” observed in 7~-p + won. 4.
nfp --f wA++
The reaction is dominated by p exchange. According to arguments that were presented before there is one unit of helicity change at the pA++ vertex, and another unit of helicity change at the meson vertex. At high enough energy K4B 2 15 GeV/c)(du/dt) will have the form -(du/dt) w Jo2(R d-t’)
+ JzZ(R 4-t’).
It has a forward peak followed by a local minimum dip should be found near -t’ N 0.6 GeV2/c2.
at -t’
- 0.1 GeV2/c2. No
10
DAR
3.07 GeV/c
IL-
A
F % A
0.01 6 0.001
I
\
3.67 GeV/c
13.3 GeV/c
\
-t [G&/d* FIG.
1. Calculation
of du/dt compared to data [2] for r-p -+ non.
I
HIGH
ENERGY
EXCHANGE
REACTIONS
+c
l= k
a 0 k t a b
c +t
+c k
k
c ‘k
t cl ‘k
01
I
4
0.2
FIG. 6. Calculation GeV.
0
0.1
I
-t
0.6
GeV2/c2
0.8
1.0
1.2
1.4
1.6
of do/dt for yp + WON* and “/p --+ ,,oN*+ at
0.4
~~~~ 7. Calculation
I
I .4
I .6 .8 -t’[GeV/c12
I
GeV/c
I 1.0
I 1.2
1.4
of da/dt compared to data [6] for n+p- p+p.
.2
8
FIG. 8. Calculation d-p + UN*++.
T+P
of da/dt
to data
3 - 4 GeV/c
compared
-t’[GeV/cl’
-UN%++
[6, 71 for
I 0.1
I I I I .I5 .20 .30 .25 -t’GeV2/c2 FIG. 9. Calculation of du/dt compared to data [5] for n-p + p%.
I .05
11.2 GeV
lT-p -poll
HIGH
ENERGY
EXCHANGE
REACTIONS
15
16
DAR QUANTITATIVE
PREDICTIONS
Quantitative predictions require more specific assumptions. A set of such assumptions is offered by the peripheral model described in Ref. [26]. Indeed this model reproduces all our qualitative predictions. This is demonstrated in Figs. l-10, where we present detailed predictions for some of the reactions that are listed in Tables I and II. The calculations are described in greater detail in Ref. [26]. The coupling constants that were used are listed in Table IV. Note in particular that (a) the calculations reproduce quite accurately the observed structure in do/d& (b) no parameters were fitted directly to the experimental data! TAF3LE JS’ Coupling constants Coupling constants g&,/4?r = 2.6 v -- itETP9m gpm
Source and remarks F&+2~)=130MeV Universality [31]
Coupling
[22] g,,,,v = -gwnv/ fi g only = -bbyP d/J &py
=
g zoy
Vector meson-photon analogy [3 1 ] qu.8 -+ 3a) = r(w+p?r+3?r) = 12 MeV [32] Isospin invariance g;,,/4r
= 0.038
gpy
=
v glu%w
= 3 g&9 = 4.12
d,,l&
Q&my
r(u-+‘y)
constants
=
G$,,/4a
-
If2 -
g,,,
d&q,
= 36
(G, = Gz , G, = 0)
Quark model [33] (In the calculations we used r)x mixing angles of -0.19 [34].) W(6) or vector dominance applied to N*+py Stodolsky Sakurai model
[311 g&&h
= 14.7
= 1.2MeV[32] g,,, = ~‘2 g,,, GN*,~/~T = 0.37 Universality [31] g&,/4a = 2.5 Vector meson-photon g&K,,o/4n = 0.80 analogy [31] g&+/4r = 1.61
=‘(6)
Source and remarks
Dispersion theory applied to nN scattering [35] Isospin invariance T(N*-+pn) = 120MeV[32] TCf-2~) = 95MeV [32] r(K*-+&P) = 16MeV [32] T(K*+K?r+)=32MeV[32]
Two COMMENTSON OTHER APPROACHES (i) First, we would like to draw attention to the fact that any model which is based on exchange mechanism with strong absorption of low partial waves should result in the same qualitative predictions. In particular this should be true for the strong Absorptive Regge Pole Exchange Models [29]. (ii) An alternative “dip
HIGH ENERGY EXCHANGE REACTIONS
17
mechanism” was recently proposed by Bander and Gotsman [30]. However, their mechanism (a) is inconsistent with W(3)-symmetry for the reaction vertices, and (b) assumes factorization on the one hand, but calls for important contributions from Regge Cuts on the other hand.
CONCLUSION
A simple interpretation of the dips and spikes observed in the t dependence of du/dt for high-energy exchange reactions has been proposed, which is in excellent agreement with the experimental
observations.
REFERENCES 1. For a recent review of the Regge Pole Exchange Model see G. H. HITE, Rev. Mod. Phys. 41 (1969),
669.
2. M. WAHLING et al., MIT-PISA Collaboration, Proc. Int. Conf. on High Energy Physics, Dubna 1964, A. V. STIRLING et al., Phys. Rev. Lett. 14 (1965), 763. 3. G. S. ABRAMS, Phys. Rev. Lett. 23 (1969), 673. 4. N. ARMENISE et al., OBBF Collaboration, Phys. Lett. 26B (1968), 336. 5. A. P. CONTOGOURIS et al., Phys. Rev. Lett. 19 (1967), 1352. 6. M. ADERHOLZ et al., ABC Collaboration, Nucl. Phys. B8 (1968), 45. 7. D. G. BROWN, Ph.D. Thesis, UCRL-18254, May, 1968. 8. G. C. BOLON et al., MIT-CEA Collaboration, Phys. Rev. Lett. 18 (1967), 926; M. BRAUNSCHWEIGet al., BONN-DESY Collaboration, Phys. Lett. 22 (1966), 705, Phys. Letters 26B (1968); R. ANDERSON et al., SLAC, Phys. Rev. Lett. 21 (1968), 384. 9. G. C. BOL~N et al., MIT-CEA Collaboration, preprint. 10. D. BELLENGER et al., MIT-CEA Collaboration, Phys. Rev. Lett. 21 (1968), 1205; W. BRAUNSCHWEIG et al., BONN-DESY Collaboration, DESY preprint. 11. S. M. Pauss et al., Argonne-Michigan Collaboration. 12. P. ASTBURY et al., Phys. Lett. 23 (1966), 396. 13. M. FERRO LUZZI et al., Nuovo Cimento 36 (1965), 1101; G. R. LYNCH et al., Phys. R u. Lett. 9 (1964), 359; E. BOLDT et al., Phys. Rev. 133 (1964), B220. 14. B. D. HY.QS et al., Nucl. Phys. B7 (1968), 1. 15. N. ARMENISE et al., OBBF Collaboration. 16. F. T. MEIE~ et al., Phys. Rev. Letters 24 (1970), 332. 17. N. ARMENLVE et al., BBF Collaboration. 18. A. M. BOYARSKI et al. SLAC, Phys. Rev. L&t. 20 (1968), 300,21(1968), 1767; B. BUSCHHORN et al., DESY, Phys. Rev. Lett. 17 (1966), 1027, 18 (1967), 571. 19. P. HEIDE et al., DESY, Phys. Rev. Lett. 21 (1968), 248; A. M. BOY~KI et al., SLAC, Phys. Rev. L&t. 21 (1968), 1767. 20. A. M. BOYARSKI et al., SLAC, Phys. Rev. Lett. 22 (1969), 1131. 21. A. ADERHOLZ et al., ABCLV Collaboration, Nucl. Phys. B5 (1968), 567, B7 (1968), 111; S. C. BERLINGHIERI et al., Rochester, Nucl. Phys. B8 (1968), 333; D. D. CARMONY et al., Nucl. Phys. B12 (1969), 9.
18
DAR
22. S. C. BERLMGHIERI et al., Ref. [21]. Y. GOLDSCHM~DT-CLERMONT, private communication. 23. A. DAR, Proceedings of the Third Int. Conf. on High Energy Physics and Nuclear Structure, Columbia University, New York, 1969. See also H. HARARI, Proceedings of the IV Int. Conf. on Interactions of Electrons and Photons at High Energies, Liverpool, 1969. 24. A. DAR AND W. TOBOCMAN, Phys. Rev. Lett. 12 (1964), 511. 25. H. HARARI, Phys. Rev. Lett. 24 (1970), 286. 26. A. DAR, T. L. WA?TS AND V. F. WEISSKOPF,Nucl. Phys. B13 (1969), 477. 27. A similar and more detailed study of forward structure and its connection to long-range pion exchange has been made also by P. R. STEVENS, UCLA preprint. 28. L. DURAND, III AND Y. T. CHIU, Phys. Rev. 139 (1965), B646; K. GOTTFRIED AND J. D. JACKSON, Nuovo Cimento 34 (1965), 735. 29. For a general review see, for instance, J. D. JACKSON, Proceeding of “The Lund Int. Conf. on High Energy Physics,” Lund, July 1969. 30. M. BANKER AND E. GOTSMAN, preprint, University of California, Irvine. 31. J. J. SAKURAI, Ann. Phys. (N. Y.) 11 (1960), 1; M. GELL-MANN AND F. ZACHARIASEN, Phys. Rev. 124 (1961), 953; L. STODOLSKY,Phys. Rev. 134B (1964), 1099; J. J. SAKURAI, Phys. Rev. Lett. 17 (1966), 1021; M. K. BANERGIE AND C. A. LEVINSON, Phys. Rev. 176 (1968), 2140. 32. Particle data group, Rev. Mod. Phys. 40 (1968), 77. 33. See, for instance, A. DAR AND V. F. WEISSKOPF,Phys. Len. 26B (1968), 670. 34. R. H. DALITZ ANLJ G. SUTHERLAND, Nuovo Cimento 37 (1965), 1777,38 (1965), 1945. 35. V. K. SAMARANAYAKE AND W. S. WOOLCOCK, Phys. Rev. Lett. 15 (1945), 936.
OF PHYSICS:
69, l-18 (1972)
Dips and Spikes in du/dt for High Energy Exchange Reactions* ARNON DAR Physics
Department,
Technion-Israel
Institute
of Technology,
Haifa,
Israel
Received August 3 1, 1970
A simple interpretation is offered for the origin of dips and spikes in do/dt for highenergy meson exchange reactions. Quantitative predictions are in good agreement with experimental measurements.
The differential cross section for two-body and quasi two-body final states in high-energy two-body collisions often exhibits a significant structure as function oft. In particular much attention has been focused on the dip at -t - 0.6 GeV2/c2 that was found in some reactions dominated by the even-signature vector meson exchange and on the dramatic narrow structure that was found in the small -t region, --t 22 mn2, of reactions dominated by w exchange. Naturally, many attempts have been made to understand these structures in terms of Regge Pole Exchange (RPE) phenomenology [l]: The dip at -t - 0.6 GeV2/c2 has normally been associated in the RPE model with the vanishing of the residues of the oddsignature vector meson trajectories at the nonsense wrong-signature point corresponding to the above t value. However, the seemingly erratic presence or absence of this dip (see Table I) has cast doubt on this explanation. Moreover, the simple Regge Pole Exchange Model with factorization also failed completely to explain the observed narrow forward structure in the cross sections for reactions dominated by 7~exchange. Some reactions which exhibit this narrow structure are listed in Table II. In this paper I would like to discuss a simple interpretation of the observed t dependence of cross sections for high-energy exchange reactions. A preliminary report on this work was included in a previous publication [23]. Here I will attempt to reformulate our basic assumptions in terms of general principles *Invited talk presented at the V-th “Recontre De Moriond,” Meribel-Les-Allues, France, March 9-20, 1970. Submitted for the Amos de4halit Memorial Volumes of Annals of Physics, Volumes 63 and 66, 1971, this article was inadvertently omitted from these dedicatory issues. Work supported in part by Stiftung Volkswagenwerk.
1 0 1972 by Academic Press, Inc.
595/69/r-1
2
DAR TABLE Reaction n-p a+n r-p a”P +P m+p +p
--f + +
7&l wp wn POP P*P WA++ +‘A++
YP - “““P yn + ?I% YP - TOP dp - K+Z+ K-p + Ron K+p - K”A+f
Dominant
exchange Dip at -i - 0.6 GeV2/c2 ? Reference
n-p --t p% 7T+n + pop n-P + Pir” a +P - P*P r+p -+ pod++ ~+p ---f pfrA++ dp dfoA++ a+n +fOp yp + n+n yn -+ r-p yp -n-A++ yn + +AK*p + K**p K-p --, K*n Kfp + K*A
Yes No No Yes Yes No Yes Yes Yes No Yes No No
2 3 4 5 6 7 7 8 9 10 11 12 13
Structure at -f ’ < m,,=
Reference
P P P 0 (TbJ P P w w
K”&O) P + P +
A 4
TABLE Reaction
I
Dominant
exchange
II
Dip Dip Spike Dip Spike Dip Spike Dip Spike Spike Dip Dip Dip Dip Spike
14 15 16 6 6,7 6 6 17 18 19 20 20 21 21 22
a ptr denotes p transversely polarized in the helicity frame.
rather than in terms of a specific peripheral model. I will then discuss a few representative reactions in some detail. A detailed comparison between theoretical predictions and a representative set of experimental results will be given. Finally, some comments regarding other approaches will be made.
3
HIGH ENERGY EXCHANGE REACTIONS
BASIC ASSUMPTIONS Our model is based on two main assumptions: (i) Exchange Reactions are “Surface Reactions” [24]; The S-channel partial waves which give the dominant contribution to an exchange reaction are the peripheral partial waves corresponding to s-channel resonances lying on an effective Regge trajectory j + l/2 - kR, where R is the hadronic radius and it increases only logarithmically with s, and k is the c.m. momentum in the S channel. This follows from Regge behaviour + duality [25]. This picture is also consistent with that underlying the absorption models, where the contribution of partial waves with j + l/2 < kR is strongly suppressed due to competition of many open channels. (ii) The peripheral partial waves of an exchange amplitude are approximately described by the corresponding partial waves of the pole closest to the physical region (the corresponding partial waves of the Born approximation for an exchange of the lightest particle lying on the exchanged Regge trajectory). This follows from the assumed analytical properties of exchange amplitudes together with the dispersion theory.
THE ORIGIN OF DIPS AND SPIKESIN du/dt FOR EXCHANGE
REACTIONS
What are the implications of (i) and (ii) for high-energy exchange reactions ? To answer this question let us examine the impact parameter expansion of an S-channel helicity amplitude for a reaction a + b ---f c + d with the helicity situation [h]; [A] = [A,, Aa, A, , A,]. It is given by FM = k2 fm b db J,,(b +t’)h(b). 0
h,+](b) is the contribution at impact parameter b to the scattering amplitude FI,] . J&b l/-t’), the cylindrical Bessel function of order dh, is a small angle and large j approximation for the rotation functions dL,(cosO), j being related to the impact parameter b through the classical relation j + l/2 - kb. p = A, - Xb,
and AX= t.~ - v is the total helicity change in the reaction. = t - tmin , where tmin is the minimum momentum transfer allowed by the kinematics of the reaction. According to (i),&(b) is appreciable only for b 2 R, i.e., v=X,-Ad t'
h](b)
s”;; 0.
(2)
4
DAR
According to (ii),
where Bl,] is the Born-approximation expression for the contribution of impact parameter b to the s-channel helicity amplitude with the helicity situation [h]. &l(b) for the exchange of a particle with mass m, and spin J, is given by
where (pe2 = m, 2 - tmin). Kdh is the cylindrical Bessel function of second kind and of order dA. It approaches rather quickly its asymptotic behaviour K&) &I
in the high-energy limit
1/( 7f/2x) e-=.
g&
(5)
depends only on the helicity situation,
but not on
b or on s.
When expressions (2), (3) and (4) are inserted in expression (1) they yield [26] hl
-
crAlw&JJ”-l
-
C~Al(s/s,-,)J6 dm
k
2
L,
b db JAb d-0
Kdb.4, (6)
e-“eR(“‘J,,(R
d/-f’)/(p,2
-
t’).
Consequently the t dependence of daldt is given by (7)
where [h] runs over all possible helicity situations.
NARROW
FORWARD
STRUCTURE
[27]
Let us first examine the behaviour of du/dt which results from TT exchange, in the forward direction. For small values of argument the cylindrical Bessel functions can be approximated by
Jk) so that da dt-
lxWA
XZA--zrY
dh positive
(8) (9)
HIGH
ENERGY
EXCHANGE
REACTIONS
5
If P,,~, the distance from the physical t region to the pion pole, is small enough, then the terms with AA = 0 on the r.h.s. of (7) when multiplied by the pion propagator will have a sharp forwardpeak, while the terms with AX # 0 when multiplied by the pion propagator will have a narrow forward dip. For high enough energy tmin -+ 0, so that P,,~ - mli2, and the type of the forward structure (a spike or a narrow forward dip, respectively) in du/dt depends only on the relative magnitudes of the different helicity amplitudes contributing to (9); i.e., on the relative magnitudes of the coefficients Cth] . These coefficients can be determined in the following way: Let us write the Born approximation for pion exchange in the form
where the Y’s are the vertex function for the corresponding It is then easy to show that the coefficients CL,,] are given by
Feynman diagram.
Table III lists the vertex functions which enter the reactions that are listed in Table II. In Table III eP is the spin-one polarization vector, e,,” is the spin-two polarization tensor, and u and U, are the spinors for the spin-112 and 312 particles. TABLE
III
cm vertex
1-o-o2+0-0-
2goazeii*@‘V p’ 4&.7? * m, e&f, QppppY
1/2+1/2+0-
&?w,
3/2+1/2+0-
$f
OYdP,
qp’h’)p’u(p,
4
#I)
We used the conventional definitions for the coupling constants, and the symbols p, p’ for the intial and final c.m. momentum, respectively. Explicit evaluation of the VA,,+in the high-energy limit p z p’ + co leads to the following rules:
(4 If I mc- maI > m, then I VA,&, m,“)l > I 6a,A,+, m,2)l for any h,h,h,‘h,‘, (b)
satisfying 1A,’ - A,’ j > 1A, - A, 1, If m, = m, and Jcp = J,’ = l/2+, then V,,, 1,2(~, m,,3 z 0.
6
DAR
These two rules are sufficient in order to correctly predict the observed forward structures that are summarized in Table II: (1)
r-p -+ pan, r+n + pop, r*p + pip, Kfp + K**p, K-p + K*n, rr+n-+fOp.
According to rule (a) the meson vertex prefers to conserve helicity, while according to rule (b) the nucleon must flip its helicity. Consequently the net helicity change is f 1, which leads to a narrow forward dip in daldt. (2)
n-p -+ p&n, yp + rr+n, yn + r-p.
There is one unit of helicity change at both the p$mr and the YTMTvertices. However, this helicity change is compensated for by helicity flip at the nucleon vertex [rule (b)]. Consequently the net helicity change is 0, which leads to a forward spike in duldt. (3)
r+p + pOA++, Kfp + K*A, rr+p + f OA++.
According to rule (a) both vertices perfer to conserve helicity. The net helicity change is therefore 0, which leads to a forward spike in du/dt. (4)
rr+p + p&A++, yp --f a-A++, yn + r+A-.
There is one unit of helicity change at both the p&mr and ynrr vertices while the baryon vertex prefers to conserve helicity [rule (a)]. The net helicity change is f 1, which leads to a narrow forward dip in dcrldt. Since experiments are performed at finite energies, the effect of tmin + 0 on the various cross sections and polarizations is worth studying. The most dramatic effect of tmin comes through pa2 in the pion propagator on the r.h.s. of (9), CL,,* being the distance of the pion pole from the physical t region. From the condition for a narrow forward structure in’daldt with a width of the order ~rn,~,
one can easily work out through the kinematics of any particular reaction the energy where one may expect narrow structure to start showing up. In particular, for reactions where mass change takes place only at one vertex, tmin --f 0 like @, while for reactions where mass change takes place at both vertices tmin -+ 0 only like s-l. We therefore expect the narrow forward structure in reactions of the first type to appear at relatively lower energies than those for reactions of second type. At lower energies t.~,~ increases, the forward structure becomes broader, and the forward cross section tends to decrease. For tmin > mlr2 no significant narrow structure is expected.
7
HIGH ENERGY EXCHANGE REACTIONS DIPS AND PEAKS IN THE INTERMEDIATE
--t
REGION
Let us now examine the intermediate --C region. The Bessel functions which appear on the r.h.s. of (7) can be recognised as the diffraction amplitude for radiation originating from a ring of radius R and with an angular momentum change Ah. These Bessel functions have simple properties as function of their argument and their order. Beyond their first maximum (around R d/--tl N Al) the cylindrical Bessel functions approach rather quickly their asymptotic form JLdx) xyA:dhmcos{x
- (Tr/2)[dh + l/2]}.
(12)
Consequently
[J&912 + [JddX)12 &
w7x)-
Since dc/dt is made of an incoherent sum of s-channel helicity amplitudes expression (7)] we may conclude from (7), (12) and (13) that
(13) [see
(a) Individual terms of the r.h.s. of (7) exhibit a diffraction pattern of maxima and minima corresponding to the extrema and zeroes, respectively, of expression (12).
(b) If a single helicity situation dominates the r.h.s. of (7), dujdt will exhibit the corresponding diffraction pattern. (c) The positions of the diffraction minima and maxima do not depend on the specific final state. They depend only on the parity of the total helicity change in the dominant amplitude on the r.h.s. of (7) [see expression (12)]. (d) The diffraction extrema are equally spaced when plotted as function of m [see expression (12)]. (e) The diffraction pattern shrinks logarithmically with increasing s, since R increases logarithmically as function of s. (f) If the helicity amplitudes with AX even and with Ah odd, respectively, contribute about equally to the r.h.s. of (7) we expect no “dip” structure in dojdt [this follows from (13)]. (g) Polarized cross sections should exhibit more structure than unpolarized cross sections. Rules (a)-(g) are the key to the “dip structure” of do/dt. We would like to stress however that all these predictions are only approximate. In general, both finite width effects and the imaginary part of the amplitude tend to wash out the diffraction pattern. In particular we do not expect any significant structure in da/dt at large -t values for reactions dominated by n exchange.
8
DAR
(Due to the small pion mass these reactions are poorly localized at a well-defined radius.) Note also that prediction (d) is in conjlict with simple RPE theory, since in the RPE models dips in du/dt are equally spaced as function of -t. We have seen by now that the presence or absence of a “dip structure” in do/dt for a particular exchange reaction depends on the relative magnitudes of all helicity amplitudes contributing to that reaction, i.e., on the relative magnitudes of the coefficients CL*] for all possible helicity situations. In the general case no simple rules can be given to determine the relative magnitudes of the coefficients CL~I , similar to the rules for pseudoscalar exchange. A general prescription for calculating these coefficients was given in Ref. [28] and [26]. The calculation in general requires (a) Knowledge of all relevant coupling constants. (b) Laborious Feynman algebra (Calculation of the Born approximation for one-particle exchange Feynman diagrams in the helicity representations and for all possible helicity situations). (c) Partial amplitudes.
wave expansion
of the s-channel Born-approximation
helicity
Many calculations of that type were reported in the literature concerned with absorption models. Below, we will analyze only a selected set of reactions for which analytic calculations are relatively simple, and for which the relevant coupling constants are known either from experimental decay rates or from reasonably successful symmetry schemes. (Generally in order to avoid tedious algebra and human mistakes, we recommend the use of the computer program that was described in Ref. [26].) 1. rr-p -+ rr% The reaction is dominated by p exchange. In the high-energy limit the ratio of the helicity flip contribution to the nonhelicity flip contribution to the r.h.s. of (7) is given by (m,2/4m,2)(GT/GV)2. From p-photon analogy [29] the ratio of the magnetic to electric coupling of the p to the nucleon is given by GTIGV = p9 - p,, - 1 = 3.7,
where pa and CL,,are the magnetic moments of p and n, respectively, in nuclear magnetons. Consequently, the term with dh = 1 in (7) dominates da/dt. For R N 0.8 fm it gives rise to diffraction dips at -t E 0,0.6,2.8, *** GeV2/c2, corresponding to the zeroes of J1(R V=?).
HIGH
2.
ENERGY
EXCHANGE
9
REACTIONS
Type I. rr+n -+ wp, r-p + wn, yp -+ q”p, Type II: rp -+ pp, yp -+ crop, yn - ran
The reactions in Type I are dominated by p exchange (note that g,,, > gw,,). The reactions in Type II are dominated by w exchange (note that gU,,” > g,,,,). For both types of reactions the helicity change in the meson vertex is f 1, because X, = f 1 and because a vector exchange couples only to helicity & 1 states of the external vector meson at the vertex. Ctnl for the four independent nonvanishing helicity amplitudes in the right-energy limit, are given by Cc,
1 l/21
1/2
C[o-
1/2
Go
112 I-
Go-
112
-
G”
+
1 1/21
-
GT
1/21
-
GT
-
GV +
I -l/z1
GT
G’
(Ah (Ah (Ah (Ah
= = = =
l), 0), 2),
1).
From p-photon analogy we know that GOT> G, v. Consequently for all reactions of Type I all CInl are about equal. da/& for these reactions should then exhibit no “dip structure”. From w-photon analogy we know further that G,” > GUT. Consequently for all reactions of Type II we expect the terms with 1Ah 1 = 1 in (7) to dominate du/dt. These terms give rise to the diffraction dips at -t’ = 0,0.6,2.8 .*. GeV2/c2 in analogy with r-p + non. Note that due to the terms with I Ah I = 2 the dip at -t’ = 0 is expected to be more pronounced, while the dip at -t’ - 0.6 GeV2/c2 is expected to be less pronounced than the corresponding dips in rr-p + non. 3.
r+p + rod++
The reaction is dominated by p exchange. p-photon analogy suggests that p is coupled to the pA vertex through a pure magnetic dipole coupling (StodolskySakurai Model [30]). This coupling favours unit helicity change at the vertex. Thus the terms with / Ah I = 1 on the r.h.s. of (7) will dominate du/dt. du/dt will then exhibit “dip structure” similar to the “dip structure” observed in 7~-p + won. 4.
nfp --f wA++
The reaction is dominated by p exchange. According to arguments that were presented before there is one unit of helicity change at the pA++ vertex, and another unit of helicity change at the meson vertex. At high enough energy K4B 2 15 GeV/c)(du/dt) will have the form -(du/dt) w Jo2(R d-t’)
+ JzZ(R 4-t’).
It has a forward peak followed by a local minimum dip should be found near -t’ N 0.6 GeV2/c2.
at -t’
- 0.1 GeV2/c2. No
10
DAR
3.07 GeV/c
IL-
A
F % A
0.01 6 0.001
I
\
3.67 GeV/c
13.3 GeV/c
\
-t [G&/d* FIG.
1. Calculation
of du/dt compared to data [2] for r-p -+ non.
I
HIGH
ENERGY
EXCHANGE
REACTIONS
+c
l= k
a 0 k t a b
c +t
+c k
k
c ‘k
t cl ‘k
01
I
4
0.2
FIG. 6. Calculation GeV.
0
0.1
I
-t
0.6
GeV2/c2
0.8
1.0
1.2
1.4
1.6
of do/dt for yp + WON* and “/p --+ ,,oN*+ at
0.4
~~~~ 7. Calculation
I
I .4
I .6 .8 -t’[GeV/c12
I
GeV/c
I 1.0
I 1.2
1.4
of da/dt compared to data [6] for n+p- p+p.
.2
8
FIG. 8. Calculation d-p + UN*++.
T+P
of da/dt
to data
3 - 4 GeV/c
compared
-t’[GeV/cl’
-UN%++
[6, 71 for
I 0.1
I I I I .I5 .20 .30 .25 -t’GeV2/c2 FIG. 9. Calculation of du/dt compared to data [5] for n-p + p%.
I .05
11.2 GeV
lT-p -poll
HIGH
ENERGY
EXCHANGE
REACTIONS
15
16
DAR QUANTITATIVE
PREDICTIONS
Quantitative predictions require more specific assumptions. A set of such assumptions is offered by the peripheral model described in Ref. [26]. Indeed this model reproduces all our qualitative predictions. This is demonstrated in Figs. l-10, where we present detailed predictions for some of the reactions that are listed in Tables I and II. The calculations are described in greater detail in Ref. [26]. The coupling constants that were used are listed in Table IV. Note in particular that (a) the calculations reproduce quite accurately the observed structure in do/d& (b) no parameters were fitted directly to the experimental data! TAF3LE JS’ Coupling constants Coupling constants g&,/4?r = 2.6 v -- itETP9m gpm
Source and remarks F&+2~)=130MeV Universality [31]
Coupling
[22] g,,,,v = -gwnv/ fi g only = -bbyP d/J &py
=
g zoy
Vector meson-photon analogy [3 1 ] qu.8 -+ 3a) = r(w+p?r+3?r) = 12 MeV [32] Isospin invariance g;,,/4r
= 0.038
gpy
=
v glu%w
= 3 g&9 = 4.12
d,,l&
Q&my
r(u-+‘y)
constants
=
G$,,/4a
-
If2 -
g,,,
d&q,
= 36
(G, = Gz , G, = 0)
Quark model [33] (In the calculations we used r)x mixing angles of -0.19 [34].) W(6) or vector dominance applied to N*+py Stodolsky Sakurai model
[311 g&&h
= 14.7
= 1.2MeV[32] g,,, = ~‘2 g,,, GN*,~/~T = 0.37 Universality [31] g&,/4a = 2.5 Vector meson-photon g&K,,o/4n = 0.80 analogy [31] g&+/4r = 1.61
=‘(6)
Source and remarks
Dispersion theory applied to nN scattering [35] Isospin invariance T(N*-+pn) = 120MeV[32] TCf-2~) = 95MeV [32] r(K*-+&P) = 16MeV [32] T(K*+K?r+)=32MeV[32]
Two COMMENTSON OTHER APPROACHES (i) First, we would like to draw attention to the fact that any model which is based on exchange mechanism with strong absorption of low partial waves should result in the same qualitative predictions. In particular this should be true for the strong Absorptive Regge Pole Exchange Models [29]. (ii) An alternative “dip
HIGH ENERGY EXCHANGE REACTIONS
17
mechanism” was recently proposed by Bander and Gotsman [30]. However, their mechanism (a) is inconsistent with W(3)-symmetry for the reaction vertices, and (b) assumes factorization on the one hand, but calls for important contributions from Regge Cuts on the other hand.
CONCLUSION
A simple interpretation of the dips and spikes observed in the t dependence of du/dt for high-energy exchange reactions has been proposed, which is in excellent agreement with the experimental
observations.
REFERENCES 1. For a recent review of the Regge Pole Exchange Model see G. H. HITE, Rev. Mod. Phys. 41 (1969),
669.
2. M. WAHLING et al., MIT-PISA Collaboration, Proc. Int. Conf. on High Energy Physics, Dubna 1964, A. V. STIRLING et al., Phys. Rev. Lett. 14 (1965), 763. 3. G. S. ABRAMS, Phys. Rev. Lett. 23 (1969), 673. 4. N. ARMENISE et al., OBBF Collaboration, Phys. Lett. 26B (1968), 336. 5. A. P. CONTOGOURIS et al., Phys. Rev. Lett. 19 (1967), 1352. 6. M. ADERHOLZ et al., ABC Collaboration, Nucl. Phys. B8 (1968), 45. 7. D. G. BROWN, Ph.D. Thesis, UCRL-18254, May, 1968. 8. G. C. BOLON et al., MIT-CEA Collaboration, Phys. Rev. Lett. 18 (1967), 926; M. BRAUNSCHWEIGet al., BONN-DESY Collaboration, Phys. Lett. 22 (1966), 705, Phys. Letters 26B (1968); R. ANDERSON et al., SLAC, Phys. Rev. Lett. 21 (1968), 384. 9. G. C. BOL~N et al., MIT-CEA Collaboration, preprint. 10. D. BELLENGER et al., MIT-CEA Collaboration, Phys. Rev. Lett. 21 (1968), 1205; W. BRAUNSCHWEIG et al., BONN-DESY Collaboration, DESY preprint. 11. S. M. Pauss et al., Argonne-Michigan Collaboration. 12. P. ASTBURY et al., Phys. Lett. 23 (1966), 396. 13. M. FERRO LUZZI et al., Nuovo Cimento 36 (1965), 1101; G. R. LYNCH et al., Phys. R u. Lett. 9 (1964), 359; E. BOLDT et al., Phys. Rev. 133 (1964), B220. 14. B. D. HY.QS et al., Nucl. Phys. B7 (1968), 1. 15. N. ARMENISE et al., OBBF Collaboration. 16. F. T. MEIE~ et al., Phys. Rev. Letters 24 (1970), 332. 17. N. ARMENLVE et al., BBF Collaboration. 18. A. M. BOYARSKI et al. SLAC, Phys. Rev. L&t. 20 (1968), 300,21(1968), 1767; B. BUSCHHORN et al., DESY, Phys. Rev. Lett. 17 (1966), 1027, 18 (1967), 571. 19. P. HEIDE et al., DESY, Phys. Rev. Lett. 21 (1968), 248; A. M. BOY~KI et al., SLAC, Phys. Rev. L&t. 21 (1968), 1767. 20. A. M. BOYARSKI et al., SLAC, Phys. Rev. Lett. 22 (1969), 1131. 21. A. ADERHOLZ et al., ABCLV Collaboration, Nucl. Phys. B5 (1968), 567, B7 (1968), 111; S. C. BERLINGHIERI et al., Rochester, Nucl. Phys. B8 (1968), 333; D. D. CARMONY et al., Nucl. Phys. B12 (1969), 9.
18
DAR
22. S. C. BERLMGHIERI et al., Ref. [21]. Y. GOLDSCHM~DT-CLERMONT, private communication. 23. A. DAR, Proceedings of the Third Int. Conf. on High Energy Physics and Nuclear Structure, Columbia University, New York, 1969. See also H. HARARI, Proceedings of the IV Int. Conf. on Interactions of Electrons and Photons at High Energies, Liverpool, 1969. 24. A. DAR AND W. TOBOCMAN, Phys. Rev. Lett. 12 (1964), 511. 25. H. HARARI, Phys. Rev. Lett. 24 (1970), 286. 26. A. DAR, T. L. WA?TS AND V. F. WEISSKOPF,Nucl. Phys. B13 (1969), 477. 27. A similar and more detailed study of forward structure and its connection to long-range pion exchange has been made also by P. R. STEVENS, UCLA preprint. 28. L. DURAND, III AND Y. T. CHIU, Phys. Rev. 139 (1965), B646; K. GOTTFRIED AND J. D. JACKSON, Nuovo Cimento 34 (1965), 735. 29. For a general review see, for instance, J. D. JACKSON, Proceeding of “The Lund Int. Conf. on High Energy Physics,” Lund, July 1969. 30. M. BANKER AND E. GOTSMAN, preprint, University of California, Irvine. 31. J. J. SAKURAI, Ann. Phys. (N. Y.) 11 (1960), 1; M. GELL-MANN AND F. ZACHARIASEN, Phys. Rev. 124 (1961), 953; L. STODOLSKY,Phys. Rev. 134B (1964), 1099; J. J. SAKURAI, Phys. Rev. Lett. 17 (1966), 1021; M. K. BANERGIE AND C. A. LEVINSON, Phys. Rev. 176 (1968), 2140. 32. Particle data group, Rev. Mod. Phys. 40 (1968), 77. 33. See, for instance, A. DAR AND V. F. WEISSKOPF,Phys. Len. 26B (1968), 670. 34. R. H. DALITZ ANLJ G. SUTHERLAND, Nuovo Cimento 37 (1965), 1777,38 (1965), 1945. 35. V. K. SAMARANAYAKE AND W. S. WOOLCOCK, Phys. Rev. Lett. 15 (1945), 936.