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Electron-positron annihilation into hadrons at high energies
Volume 41B, number 3
ELECTRON-POSITRON
PHYSICS LETTERS
ANNIHILATION
2 October 1972
INTO HADRONS AT HIGH ENERGIES
H. CHENG* Department o f Mathematics, Massachusetts Institute o f Technology, Cambridge, Massachusetts 02139, USA and T.T. WU** Gordon McKay Laboratory, Harvard University ***, Cambridge, Massachusetts 02138, USA, and Deutsches Elektronen-Synchrotron DESY, Hamburg, Germany Received 1 August 1972 In the limit of extremely high energies with the transverse momentum of each hadron fixed, the reaction e+ + e - ~ hadrons proceeds by the annihilation of this pair into two virtual photons which turn into two fireballs of hadrons with negligible final-state interaction.
Motivated by the excellent progress of the SPEAR project at SLAC, we discuss here some properties of the electron-positron interaction at extremely high energies. At extremely high energies, the dominant process for both electron-electron and electron-positron colliding beams is the so-called "two-photon process" [ 1 - 5 ] , diagramatically shown in fig. 1 (a). Since photons of spin 1 are exchanged, the total cross sections for given final states are approximately independent of energy at high energies. This dominant process, with at least two outgoing leptons, is welt-understood and will not be discussed further here. We concentrate on the process e ÷ + e - --, hadrons
of the order of x/s, while that for the process in fig. 1 (c) is of the order of sO. The process o f fig. 1 (b) is characterized by the presence of two fireballs [6, 7] of hadrons. Furthermore, each fireball is neutral. The most important question about this process is: Do the two jets interact significantly with each other at high energies? In particular, are the diagrams of fig. 2 as important as that of fig. 1 (b)? If they are, pionization products can be formed and it may not be meaningful to consider the hadrons to be segregated into two fireballs.
}
(1)
for given final state and given momentum transfers in the limit s ~ o,,, where s is as usual the square of the cm energy. For (1), the dominant processes are the ones in which an electron is exchanged. The simplest of such diagramsis shown in fig. 1 (b). Again, two virtual photons are involved. Even though this is a fourthorder process, it is more important than the secondorder process of fig. 1 (c) in the limit of interest. This is because the amplitude for the process in fig. 1 (b) is * Work supported in part by the National Science Foundation under Grant No. GP-29462. ** Work supported in part by the U.S. Atomic Energy Commission under contract No. AT (11-1)-3227. *** Permanent address.
hadrons or leptons
• ~ (a)
•~
} hadrons hadrons
(b)
(el Fig. 1. Three Feynman diagrams of decreasing importance in the limit s ~ ~ with fixed momentum transfers. 375
Volume 41B, number 3
•~ ~
PHYSICS LETTERS
hodrons }
hodrons (c)
(al
e~
} hodron$
(b)
[d)
Fig. 2. Some further diagrams for electron-positron annihilation. This question can be most easily answered by applying the impact picture [8] originally devised for hadronic interactions. For the sake of definiteness, consider e+ + e -
-> (~+ + ~r-) + (~+ + ~ - )
Many properties follow from this lack of finalstate interaction between the two fireballs. With reference to fig. 1 (b), the matrix element, in the limit of high energies, is in the form of a sum (over the induces due to the virtual photons) of products of three factors: one depends only on the variables of one fireball, a second on those of the other fireball, and a third comes from quantum electrodynamics (which is completely understood)*. In particular, all dependence on the momentum transfer, i.e., the momentum carried by the exchanged electron, is exhibited in this third factor. We give a few more simple examples of such processes in addition to (2): e + + e - ~ ( K + + K - ) +(rr + + rr-),
(3)
e + + e - -->(K0 + ~ 0 ) +(Tr+ + rr-),
(4)
e+ + e - ~ ( K + + K - ) + ( K + + K - ) ,
(5)
e+ + e - -->(K+ + K - ) + (K 0 + ~0),
(6)
e + + e - --->(2rr+ + 21r-) + (lr + + rr-),
(7)
e + + e - ~ ( K + + K - ) +(rr + + r r - + rr0),
(8)
e + + e - -->(K0 + ~ 0 ) +(rr+ + ; r - +nO),
(9)
(2)
as shown in fig. 2 (c). Here the parentheses are used to designate members of the two fireballs. Let us designate the longitudinal momenta of the four particles to rebe ½ / 3 ~ , ~ (l-/})w~, - ~/3 ' ' ,vrs-, and ~' (I-/3)V~-, ' spectively. If/3 and/3' are fixed at values between 0 and I, and the transverse momenta are also given, then the masses of the two virtual photons in fig. 2 (c) are finite, i.e., of the order s O. Therefore, these virtual photons have a finite lifetime in their respective rest system; by time dilation, their lifetimes are of the order of,v~-in the e+e - cm system. Because of these long lifetimes at extremely high energies, the two sets of pions are well separated in space by the time they are formed. Thus, at infinite energy there is no interaction between the two fireballs. Alternatively, we say that the final-state interaction between the two fireballs is less and less important at higher and higher energies. This simple, yet important and startling, point is the basis of the entire discussion here. It is thus necessary to verify it more directly. For this purpose, we study the diagram of fig. 2 (d) and compare it with that o f fig. 2 (c). The verification is complicated by the smallness of the electron mass and will be presented elsewhere. 376
2 October 1972
e+ + e - --" V + (rr+ + 7r-), and
( I O)
e++e--+V+V
(ii)
',
where V and V ' are two vector mesons, i.e., p0, co, ~, or any higher vector meson. The processes (10) and (11) [or e + + e - -~ V + 3'] may be useful in searching for additional vector mesons o f narrow width. We list below some of the physical consequences of the above-mentioned lack of final-state interaction between the two fireballs. These consequences are of course also experimental tests of the impact picture. (A) At high energies, electron-positron annihilation into hadrons is qualitatively different from, and in * In other words, the form of the matrix e l e m e n t is similar
to that for e+ + e - ~ 3"+ 3' or e+ + e - --) 3"+ hadrons. For the calculation of differential cross sections, it should be remembered that beams in electron-positron storage rings are significantly polarized.
Volume 41B, number 3
PHYSICS LETTERS
do[e + + e - . . + p + + p - ] do [e + + e - - + p O +pO]
0
(a)
2 October 1972
~'0.
(14)
(E) Each of the fireballs has spin 1. Thus, for ex-
ample do[e + + e - -~ p0 + lr0] do[e + + e - ~ p O +pO] --> O.
0 Fig. 3. Comparison of one-particle distributions for given process at high energies: (a) for electron-positron annihilation and (b) for hadron-hadron interaction.
some ways simpler than, the scattering of two hadrons [9]. In particular, for the limit under consideration, the absence of fig. 2 (b) implies that the two fireballs are well defined. (B) The absence of fig. 2 (b) also implies that pionization does not occur [ 1, 10] for electron-positron annihilation into hadrons. More precisely, for a given process such as (2), consider the one-particle distribution for either ~r+ or lr-. Let the transverse momentum p± for this particle be fixed, then a plot of this one-particle distribution against the rapidity variable 0 = sinh-lpu/~ is expected to show two peaks with very small values in-between. For higher and higher energies, the peaks are further and further apart. This situation is schematically shown in fig. 3 (a). It is to be contrasted with the case p + p -~ p + p + rr+ + rr-, for example, where the corresponding plot is quite flat, as shown schematically in fig. 4 ( b ) [1] * (C) The absence of pionization means that the hypothesis of limiting fragmentation [ 11] applies better to electron-positron annihilation into hadrons. The smallness of the electron mass must be properly taken into account. (D) Each of the fireballs is neutral. Thus, for example do[e + + e - -~ Or÷ + ~r÷) + ( ~ - + 7r-)] do [e + + e - - ~ 0 r + + rr-) + Or+ + 7r-)] ~ 0,
(12)
do[e + + e - - (~r+ + . % + Or- + "%1 do [e+ + e - - 0 r + + rr-) +(Tr+ + 7r-)] " 0, and * This has not yet been verified experimentally.
(13)
(15)
(F) Some results follow from either D or E. Thus do [e + + e - -~ n + + lr- ] do[e + + e - -+ 00 + p0] -+ 0.
(16)
(G) Each of the fireballs has strangeness O, Thus, for example do[e + + e - -~ (K + + rr-) + ( K - + rr+)] do[e + + e - ~ ( K + + K - ) + (Tr+ + rr-)]
-*0.
(17)
(H) Each of the fireballs has baryon number 0. Thus, for example do[e + + e - ~ (p + 7r-) + ( p + n+)] do[e+ + e_ ~ (p + ~) + (n+ + lr_)L] ~ 0.
(18)
It is not difficult to write down many more similar selection rules. (I) Let us give one example of a slightly more sophisticated relation. Suppose one fireball consists of only K + + K - , then they must be in a p state. Thus, the total isotopic spin is 0. Accordingly dole + + e - --> (K 0 +•0) + X] do [e + + e - ~ (K + + K - ) + X] ~ 1,
(19)
where X is any admissible fireball [and may be 7 itself]. In writing down this relation, we have neglected K + - K 0 mass difference. Eq. (19) applies in particular to (5) and (6), (8) and (9). In this connection, the recentdiscussion on KOK,0 system [12] is of interest. (J) Since the total cross section due to the onephoton process of fig. 1 (c) is believed to behave [13] as s - 1 or less for large s, the multiplicity for the process under consideration [of fig. 1 (b)] is finite as s ~ o~. This is again qualitatively different from the case of hadron-hadron interaction. Some o f the complications of the hadronic case [14] thus do not occur here. We cannot discuss precisely how high the energy 37.7
Volume 41B, number 3
PHYSICS LETTERS
large angles. Similarly, some of the Class III processes here are important for large m o m e n t u m transfers.
e3
Finally, we remark that a part of the considerations here can be extended to the problem of lepton production in hadron scattering, e.g.,
Ctoss I
Class II
p + p -~ e+ + e - + hadrons and
CIQsslII
p + p -+/a+ + / a - + hadrons.
(o)
et
e3
2tQss
I
2 October 1972
None
(20)
At very high energies, these proceed via two virtual photons. Details of this extension will be published elsewhere.
None
I Clossn
None
---~
}h - ~ h h ] ~ h + ~
h-
CI~zssgI
We are greatly indebted to Professor C.N. Yang for the most illuminating discussions. One of us (TTW) also wishes to thank Professor E. Lohrmann and Professor W. Paul for useful information about storage rings.
(b)
Fig. 4. Classification of electron-positron processes according to order and to high-energy behavior for fixed momentum transfers. The diagrams are meant to be representative, not exhaustive. (a) Electrodynamic processes. The produced particles may be electrons, muons, higher leptons, or any other charged particle without strong interactions. (b) Processes with hadron production. Here h+(h-) means a system of one or more hadrons, even (odd) under charge conjugation.
has to be for the present two-photon process to dominate over the one-photon process. This is due mainly to our lack of reliable knowledge about the one-photon process, as discussed by Bjorken and Brodsky [15]. We present in fig. 4 a simple, but perhaps useful, diagram of various electron-positron processes. The term "Class I" refers to those processes which give a roughly constant cross section for s ~ oo, even when transverse m o m e n t u m transfers are limited. Matrix elements of "Class II" processes are smaller than those of "Class I" by a factor ofs -'A, and those of "Class III" are still smaller by another factor of s -1/2. We recall the case of electron Compton scattering, where the second-order diagrams leading to the K l e i n - N i s h i n a formula are unimportant compared with the sixthorder Delbrfick-like diagrams for fLxed m o m e n t u m transfer in the limit s ~ oo [16], but are dominant for
378
References [1] H. Cheng and T.T. Wu, Phys. Rev. Lett. 23 (1969) 1311. [2] S.J. Brodsky, T. Kinoshita and H. Terazawa, Phys. Rev. Lett. 25 (1970) 972. [3] A. Jaccarini et al., Nuovo Cimento Letters 4 (1970) 933; Phys. Rev. D3 (1971) 1569. [4] H. Cheng and T.T. Wu, Phys. Lett. 36B (1971) 241; Nucl. Phys. B32 (1971) 461. [5] S.J. Brodsky, T. Kinoshita and H. Terazawa, Phys. Rev. D4 (1971) 1532. [6] G. Cocconi, Phys. Rev. 111 (1958) 1699; K. Niu, Nuovo Cimento 10 (1958) 994; P. Cork et al., ibid 10 (1958) 741. [7] R.K. Adair, Phys. Rev. 172 (1968) 1370. [8] H. Cheng and T.T. Wu, Phys. Rev. Lett. 23 (1969) 670. [9] H. Cheng and T.T. Wu, Phys. Rev. Lett. 24 (1970) 1456. [10] H. Chang and T.T. Wu, Phys. Rev. D3 (1971) 2195. [11] J. Benecke et aL, Phys. Rev. 188 (1969) 2159. [12] M. Goldhaber and C.N. Yang, in: Evolution of Particle Physics, ed. M. Conversi (Academic Press, New York, 1970), p. 171. [13] V.N. Gribov, B.L. Ioffe and I.Ya. Pomeronehuk, Phys. Lett. 24B (1967) 554. [14] C. Quigg, J.M. Wang and C.N. Yang, Phys. Rev. Lett. 28 (1972) 1290. [15] J.D. Bjorken and S.J. Brodsky, Phys. Rev. D1 (1970) 1416. [16] S.L. Wu, doctoral dissertation, Harvard University (June 1970) Chapter I.
ELECTRON-POSITRON
PHYSICS LETTERS
ANNIHILATION
2 October 1972
INTO HADRONS AT HIGH ENERGIES
H. CHENG* Department o f Mathematics, Massachusetts Institute o f Technology, Cambridge, Massachusetts 02139, USA and T.T. WU** Gordon McKay Laboratory, Harvard University ***, Cambridge, Massachusetts 02138, USA, and Deutsches Elektronen-Synchrotron DESY, Hamburg, Germany Received 1 August 1972 In the limit of extremely high energies with the transverse momentum of each hadron fixed, the reaction e+ + e - ~ hadrons proceeds by the annihilation of this pair into two virtual photons which turn into two fireballs of hadrons with negligible final-state interaction.
Motivated by the excellent progress of the SPEAR project at SLAC, we discuss here some properties of the electron-positron interaction at extremely high energies. At extremely high energies, the dominant process for both electron-electron and electron-positron colliding beams is the so-called "two-photon process" [ 1 - 5 ] , diagramatically shown in fig. 1 (a). Since photons of spin 1 are exchanged, the total cross sections for given final states are approximately independent of energy at high energies. This dominant process, with at least two outgoing leptons, is welt-understood and will not be discussed further here. We concentrate on the process e ÷ + e - --, hadrons
of the order of x/s, while that for the process in fig. 1 (c) is of the order of sO. The process o f fig. 1 (b) is characterized by the presence of two fireballs [6, 7] of hadrons. Furthermore, each fireball is neutral. The most important question about this process is: Do the two jets interact significantly with each other at high energies? In particular, are the diagrams of fig. 2 as important as that of fig. 1 (b)? If they are, pionization products can be formed and it may not be meaningful to consider the hadrons to be segregated into two fireballs.
}
(1)
for given final state and given momentum transfers in the limit s ~ o,,, where s is as usual the square of the cm energy. For (1), the dominant processes are the ones in which an electron is exchanged. The simplest of such diagramsis shown in fig. 1 (b). Again, two virtual photons are involved. Even though this is a fourthorder process, it is more important than the secondorder process of fig. 1 (c) in the limit of interest. This is because the amplitude for the process in fig. 1 (b) is * Work supported in part by the National Science Foundation under Grant No. GP-29462. ** Work supported in part by the U.S. Atomic Energy Commission under contract No. AT (11-1)-3227. *** Permanent address.
hadrons or leptons
• ~ (a)
•~
} hadrons hadrons
(b)
(el Fig. 1. Three Feynman diagrams of decreasing importance in the limit s ~ ~ with fixed momentum transfers. 375
Volume 41B, number 3
•~ ~
PHYSICS LETTERS
hodrons }
hodrons (c)
(al
e~
} hodron$
(b)
[d)
Fig. 2. Some further diagrams for electron-positron annihilation. This question can be most easily answered by applying the impact picture [8] originally devised for hadronic interactions. For the sake of definiteness, consider e+ + e -
-> (~+ + ~r-) + (~+ + ~ - )
Many properties follow from this lack of finalstate interaction between the two fireballs. With reference to fig. 1 (b), the matrix element, in the limit of high energies, is in the form of a sum (over the induces due to the virtual photons) of products of three factors: one depends only on the variables of one fireball, a second on those of the other fireball, and a third comes from quantum electrodynamics (which is completely understood)*. In particular, all dependence on the momentum transfer, i.e., the momentum carried by the exchanged electron, is exhibited in this third factor. We give a few more simple examples of such processes in addition to (2): e + + e - ~ ( K + + K - ) +(rr + + rr-),
(3)
e + + e - -->(K0 + ~ 0 ) +(Tr+ + rr-),
(4)
e+ + e - ~ ( K + + K - ) + ( K + + K - ) ,
(5)
e+ + e - -->(K+ + K - ) + (K 0 + ~0),
(6)
e + + e - --->(2rr+ + 21r-) + (lr + + rr-),
(7)
e + + e - ~ ( K + + K - ) +(rr + + r r - + rr0),
(8)
e + + e - -->(K0 + ~ 0 ) +(rr+ + ; r - +nO),
(9)
(2)
as shown in fig. 2 (c). Here the parentheses are used to designate members of the two fireballs. Let us designate the longitudinal momenta of the four particles to rebe ½ / 3 ~ , ~ (l-/})w~, - ~/3 ' ' ,vrs-, and ~' (I-/3)V~-, ' spectively. If/3 and/3' are fixed at values between 0 and I, and the transverse momenta are also given, then the masses of the two virtual photons in fig. 2 (c) are finite, i.e., of the order s O. Therefore, these virtual photons have a finite lifetime in their respective rest system; by time dilation, their lifetimes are of the order of,v~-in the e+e - cm system. Because of these long lifetimes at extremely high energies, the two sets of pions are well separated in space by the time they are formed. Thus, at infinite energy there is no interaction between the two fireballs. Alternatively, we say that the final-state interaction between the two fireballs is less and less important at higher and higher energies. This simple, yet important and startling, point is the basis of the entire discussion here. It is thus necessary to verify it more directly. For this purpose, we study the diagram of fig. 2 (d) and compare it with that o f fig. 2 (c). The verification is complicated by the smallness of the electron mass and will be presented elsewhere. 376
2 October 1972
e+ + e - --" V + (rr+ + 7r-), and
( I O)
e++e--+V+V
(ii)
',
where V and V ' are two vector mesons, i.e., p0, co, ~, or any higher vector meson. The processes (10) and (11) [or e + + e - -~ V + 3'] may be useful in searching for additional vector mesons o f narrow width. We list below some of the physical consequences of the above-mentioned lack of final-state interaction between the two fireballs. These consequences are of course also experimental tests of the impact picture. (A) At high energies, electron-positron annihilation into hadrons is qualitatively different from, and in * In other words, the form of the matrix e l e m e n t is similar
to that for e+ + e - ~ 3"+ 3' or e+ + e - --) 3"+ hadrons. For the calculation of differential cross sections, it should be remembered that beams in electron-positron storage rings are significantly polarized.
Volume 41B, number 3
PHYSICS LETTERS
do[e + + e - . . + p + + p - ] do [e + + e - - + p O +pO]
0
(a)
2 October 1972
~'0.
(14)
(E) Each of the fireballs has spin 1. Thus, for ex-
ample do[e + + e - -~ p0 + lr0] do[e + + e - ~ p O +pO] --> O.
0 Fig. 3. Comparison of one-particle distributions for given process at high energies: (a) for electron-positron annihilation and (b) for hadron-hadron interaction.
some ways simpler than, the scattering of two hadrons [9]. In particular, for the limit under consideration, the absence of fig. 2 (b) implies that the two fireballs are well defined. (B) The absence of fig. 2 (b) also implies that pionization does not occur [ 1, 10] for electron-positron annihilation into hadrons. More precisely, for a given process such as (2), consider the one-particle distribution for either ~r+ or lr-. Let the transverse momentum p± for this particle be fixed, then a plot of this one-particle distribution against the rapidity variable 0 = sinh-lpu/~ is expected to show two peaks with very small values in-between. For higher and higher energies, the peaks are further and further apart. This situation is schematically shown in fig. 3 (a). It is to be contrasted with the case p + p -~ p + p + rr+ + rr-, for example, where the corresponding plot is quite flat, as shown schematically in fig. 4 ( b ) [1] * (C) The absence of pionization means that the hypothesis of limiting fragmentation [ 11] applies better to electron-positron annihilation into hadrons. The smallness of the electron mass must be properly taken into account. (D) Each of the fireballs is neutral. Thus, for example do[e + + e - -~ Or÷ + ~r÷) + ( ~ - + 7r-)] do [e + + e - - ~ 0 r + + rr-) + Or+ + 7r-)] ~ 0,
(12)
do[e + + e - - (~r+ + . % + Or- + "%1 do [e+ + e - - 0 r + + rr-) +(Tr+ + 7r-)] " 0, and * This has not yet been verified experimentally.
(13)
(15)
(F) Some results follow from either D or E. Thus do [e + + e - -~ n + + lr- ] do[e + + e - -+ 00 + p0] -+ 0.
(16)
(G) Each of the fireballs has strangeness O, Thus, for example do[e + + e - -~ (K + + rr-) + ( K - + rr+)] do[e + + e - ~ ( K + + K - ) + (Tr+ + rr-)]
-*0.
(17)
(H) Each of the fireballs has baryon number 0. Thus, for example do[e + + e - ~ (p + 7r-) + ( p + n+)] do[e+ + e_ ~ (p + ~) + (n+ + lr_)L] ~ 0.
(18)
It is not difficult to write down many more similar selection rules. (I) Let us give one example of a slightly more sophisticated relation. Suppose one fireball consists of only K + + K - , then they must be in a p state. Thus, the total isotopic spin is 0. Accordingly dole + + e - --> (K 0 +•0) + X] do [e + + e - ~ (K + + K - ) + X] ~ 1,
(19)
where X is any admissible fireball [and may be 7 itself]. In writing down this relation, we have neglected K + - K 0 mass difference. Eq. (19) applies in particular to (5) and (6), (8) and (9). In this connection, the recentdiscussion on KOK,0 system [12] is of interest. (J) Since the total cross section due to the onephoton process of fig. 1 (c) is believed to behave [13] as s - 1 or less for large s, the multiplicity for the process under consideration [of fig. 1 (b)] is finite as s ~ o~. This is again qualitatively different from the case of hadron-hadron interaction. Some o f the complications of the hadronic case [14] thus do not occur here. We cannot discuss precisely how high the energy 37.7
Volume 41B, number 3
PHYSICS LETTERS
large angles. Similarly, some of the Class III processes here are important for large m o m e n t u m transfers.
e3
Finally, we remark that a part of the considerations here can be extended to the problem of lepton production in hadron scattering, e.g.,
Ctoss I
Class II
p + p -~ e+ + e - + hadrons and
CIQsslII
p + p -+/a+ + / a - + hadrons.
(o)
et
e3
2tQss
I
2 October 1972
None
(20)
At very high energies, these proceed via two virtual photons. Details of this extension will be published elsewhere.
None
I Clossn
None
---~
}h - ~ h h ] ~ h + ~
h-
CI~zssgI
We are greatly indebted to Professor C.N. Yang for the most illuminating discussions. One of us (TTW) also wishes to thank Professor E. Lohrmann and Professor W. Paul for useful information about storage rings.
(b)
Fig. 4. Classification of electron-positron processes according to order and to high-energy behavior for fixed momentum transfers. The diagrams are meant to be representative, not exhaustive. (a) Electrodynamic processes. The produced particles may be electrons, muons, higher leptons, or any other charged particle without strong interactions. (b) Processes with hadron production. Here h+(h-) means a system of one or more hadrons, even (odd) under charge conjugation.
has to be for the present two-photon process to dominate over the one-photon process. This is due mainly to our lack of reliable knowledge about the one-photon process, as discussed by Bjorken and Brodsky [15]. We present in fig. 4 a simple, but perhaps useful, diagram of various electron-positron processes. The term "Class I" refers to those processes which give a roughly constant cross section for s ~ oo, even when transverse m o m e n t u m transfers are limited. Matrix elements of "Class II" processes are smaller than those of "Class I" by a factor ofs -'A, and those of "Class III" are still smaller by another factor of s -1/2. We recall the case of electron Compton scattering, where the second-order diagrams leading to the K l e i n - N i s h i n a formula are unimportant compared with the sixthorder Delbrfick-like diagrams for fLxed m o m e n t u m transfer in the limit s ~ oo [16], but are dominant for
378
References [1] H. Cheng and T.T. Wu, Phys. Rev. Lett. 23 (1969) 1311. [2] S.J. Brodsky, T. Kinoshita and H. Terazawa, Phys. Rev. Lett. 25 (1970) 972. [3] A. Jaccarini et al., Nuovo Cimento Letters 4 (1970) 933; Phys. Rev. D3 (1971) 1569. [4] H. Cheng and T.T. Wu, Phys. Lett. 36B (1971) 241; Nucl. Phys. B32 (1971) 461. [5] S.J. Brodsky, T. Kinoshita and H. Terazawa, Phys. Rev. D4 (1971) 1532. [6] G. Cocconi, Phys. Rev. 111 (1958) 1699; K. Niu, Nuovo Cimento 10 (1958) 994; P. Cork et al., ibid 10 (1958) 741. [7] R.K. Adair, Phys. Rev. 172 (1968) 1370. [8] H. Cheng and T.T. Wu, Phys. Rev. Lett. 23 (1969) 670. [9] H. Cheng and T.T. Wu, Phys. Rev. Lett. 24 (1970) 1456. [10] H. Chang and T.T. Wu, Phys. Rev. D3 (1971) 2195. [11] J. Benecke et aL, Phys. Rev. 188 (1969) 2159. [12] M. Goldhaber and C.N. Yang, in: Evolution of Particle Physics, ed. M. Conversi (Academic Press, New York, 1970), p. 171. [13] V.N. Gribov, B.L. Ioffe and I.Ya. Pomeronehuk, Phys. Lett. 24B (1967) 554. [14] C. Quigg, J.M. Wang and C.N. Yang, Phys. Rev. Lett. 28 (1972) 1290. [15] J.D. Bjorken and S.J. Brodsky, Phys. Rev. D1 (1970) 1416. [16] S.L. Wu, doctoral dissertation, Harvard University (June 1970) Chapter I.