Production and Annihilation of Antiprotons Citation Helstrom, Carl Wilhelm (1951) Production and Annihilation of Antiprotons. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/RCCV-Z308. https://resolver.caltech.edu/CaltechTHESIS:10132017-101145132 Abstract If protons and neutrons are Dirac particles, as one usually assumes, the corresponding antiparticles should ex­ist, but these have never been observed. The coupling of nucleons with the meson field offers processes by which these particles could be created in energetic collisions be­tween nucleons and mesons or other nucleons. In this thesis the pseudoscalar meson theory is used to calculate cross-sec­tions for the production of antiprotons in such collision processes. These are applied to estimate the numbers of antiprotons to be expected from the interaction of cosmic­ ray particles with the nucleons of the atmosphere. It is found that meson production is about 60 times more frequent than antinucleon production for the complete primary cosmic­ ray spectrum, but that antinucleon production is of compar­able probability with meson production for energies greater than about 10 11 e.v. Cross-sections are also calculated for the annihilation of antiprotons in collisions with protons and neutrons, with emission of mesons. Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: (Physics and Mathematics) Degree Grantor: California Institute of Technology Division: Physics, Mathematics and Astronomy Major Option: Physics Minor Option: Mathematics Thesis Availability: Public (worldwide access) Research Advisor(s): Christy, Robert F. Thesis Committee: Unknown, Unknown Defense Date: 1 January 1951 Funders: Funding Agency Grant Number DuPont (United States) UNSPECIFIED Record Number: CaltechTHESIS:10132017-101145132 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:10132017-101145132 DOI: 10.7907/RCCV-Z308 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 10515 Collection: CaltechTHESIS Deposited By: Benjamin Perez Deposited On: 13 Oct 2017 18:29 Last Modified: 03 May 2023 18:41 Thesis Files Preview PDF - Final Version See Usage Policy. 25MB Repository Staff Only: item control page

PRODUCT I ON AND ANNIHILATION O!t" AN TIPROTONS Thesis by Car l Wilhelm Helstrom In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy California Institute of Technology Pasadena, California 1951 ACKNOWLEDGMENTS I wish to express my deepest appreciation to Professor R.F. Christy for suggesting this problem, which is probably an ideal one for a graduate thesis, and for so patiently guiding me in its solutiono My thanks go also to Professor R.P. Feynman and Mr. Malvin A. Ruderman for stimulating discussions of many of the points of the thesis. I am grateful to the E. r. du Pont de Nemours & Company for the generous fello wship under which this research was accomplished. ABSTRACT If protons and neutrons are Dirac particles, as one usually assumes, the corresponding antiparticles should exist, but these have never been observed. The coupling of nucleons with the meson field offers processes by which these particles could be created in energetic collisions between nucleons and mesons or other nucleons. In this thesis the pseudoscalar meson theory is used to calcula te cross-sections for the production of antiprotons in such collision processes. These are a pplied to estimate the numbers of antiprotons to be exp ected from the interaction of cosmicray particles with the nucleons of the atmosphere. It is found that meson production is about 60 times more frequent than antinucleon production for the complete primary cosmicray spectrum, but that antinucleon production is of comparable probability with meson production for energies greater than about 10 11 e.v. Cross-sections are also calculated for the annihilation of antiprotons in collisions with protons and neutrons, with emission of mesons. TABLE OF CONTENTS TITLE PART PAGE INTRODUCTION I. I I. 1 .... KINEMATICS OF THREE-PARTICLE PROCESSES. 8 - APPLICATION OF THE FEYNMAN METHOD • . 14 III. ANNIHILATION OF ANTIPROTONS • . • IV. • 2.3 PRODUCTION OF ANTIPROTON3 IN MESON-NUCLEON COLLIS IONS • . . . . . . . ., . . . . . . . . . . . . . 38 V. PRODUCTION OF ANTIPROTO NS I N NUCLEON-NUCLEON COLLISIONS-- PERTURBATION METHOD (NON-RELATIVISTIC). 51 VI. APPLICATION OF THE WEIZSACKER-WILLIAMS METHOD •• 58 VII. PRODUCTION OF MESONS IN NUCLEON-NUCLEON COLLISIONS . . . 67 VIII.PRODUCTION OF ANTIPROTONS IN COSMIC RAYS. • • 71 ...... • 75 IX. . . . CO NCLUSION. • . . . . . . • . • . . . . . Appendix I. ANNIHILATION OF ANTIPROTONS I N BOUND STATES. . . . . . . . . • . • • • • 78 Appendix II. MU- MESON DECAY SPECTRUM . . • 82 Appendix III. ADDITIONAL PROCESSES WITH ANTIPROTONS • 85 REFERENCES . • . 91 INTRODUCTION The theory of the electron proposed by Dirac in 1928 was remarkably successful in explaining the properties of that particle as known at the time 1 . For instance, it yielded the Sommerfeld expression for the fine-structure of the spectral lines of hydrogen. In addition, the fact that there was a spin angular momentum of Ml associated with the electron came automatically from the theory. However, there was an import- ant difficulty with the theory, namely that it offered the e.>cistence of states of negative energy for the electron. By in- teraction with the radiation field, transitions from states of positive to states of negative energy were possible, so that it appeared tha.t the electron could lose an infinite amount of energy, gradually coasting down through states of more and more negative energy. It was not possible to disregard these states, for their presence was necessary in order that the set of solutions of the Dirac equation be "complete" in the mathematical sense. I.e., the expansion of an arbitrary wave-func- tion in terms of the eigensolutions of the Dirac equation would in general have to include some of those associated with states of negative energy. This difficulty of negative-energy states was circumvented by the assumpti.on that in a vacuum all the states of negative energy are filled with electrons 2 • Then transitions to them are forbidden by the Pauli exclusion principle, which makes it impossible for two particles (fermions) to occupy the " -i::;- same state. But now it appeared that if sufficient energy were supplied -- namely, more than 2mc 2 , where mis the mass of the electron -- an electron in such a state of negative energy could be raised to a state of positive energy. nultant hole in the 11 The re- se a 11 of negative-energy electrons wouJd behave like a particle of electronic mass but of opposite charge. At first, Dirac associated these p ositive particles with protons, but later the discovery by Anderson and Neddermeyer of the positron pr ov ided a physical realization of these hypothetical 11 anti-particles 11 • The processes by which there are produced always involves the creation of a pair of particles, a positive and a negative electron, at the same time. Many experiments, together with considerations of nuclear and molecular structure, have demonstrated that the proton and the neutron also have a spin angular momentum of ~ and obey Fermi statistics. The only theory known which provides such properties for an elementary particle is the Dirac theory. Protons and neutrons, or "nucleons", as they are generically called, have about them the short-range nuclear force field, in addition to the electromagnetic fielci of the proton. To explain this, one assumes coupling of the nucleons with the field of mesons, so that nucleons can act as sources or sinks of mesons as well as of photons. These theories of the nuc- leon have as yet given only qualitative agreement with experiment. The lack of ex.act quantitative agreement may be due to the inadequacy of the methods of calculation. For instance, all meson theories predict that the nucleons will have an addi- -.3- tional magnetic moment due to their interaction with the meson field (emission and absorption of virtual charged mesons), but none of the theories has thus far yielded the experimental values of the anomalous moments of the proton and the neutron. A major difficulty with the interpretation of the proton as a Dirac particle is that the corresponding a nti-particle, or "antiproton 11 , should exist. This antiproton would have the same mass as the proton, but a negative charge and a magnetic moment of equal magnitude but of opposite sign. ticle has never been observed. Such a par- It is the purpose of this the- sis to investigate some of the properties predicted by current theory for the antiproton, to determine whether the fa.ct that it has not been observed can be explained in this way. The production of antiprotons through the electromagnetic interaction of protons is possible by the same type of processes whereby electron-positron pairs are made, provided sufficient energy is available. Since cross-sections for such re- actions are proportional to the square of the Compton wavelength of the particle produced, it is seen that those for making proton-antiproton pairs would be smaller by a factor of (1/1836)2 : 3 x 10- 7 than the cross-sections for producing positron-electron pairs at corresponding energies. Thus the probability of such reactions would be unobservably small. It is also through the coupling of nucleons with the mes<l.'l ]ield that we can expect reactions to occur in which nucleon pairs are produced. The coupling constant g2/tc which defines the strength of the interaction is on the order of unity, as -4- compared with the electromagnetic coupling constant E!ifric ~ 1/137. Since the coupling constant occurs to the third or fourth power in the cross-sections (computed in lowest order ) for the various production processes, this difference in the sizes of the coupling constants may offset the above noted factor of 3 x 10- 7. This theory requires that both the proton and the neutron be Dirac particles, so that there are t wo 11 sea.s", one of negative-energy protons, the other of negative-energy neutrons; and there are both antiprotons and antineutrons, each represented by "holes" in the respective seas. A neutron in a negative- energy state could give up a negative meson to become a proton which fills the hole which represents an antiproton. This cor- responds to the virtual emission of negative mesons by antipr~ tons: p-~n - + N*, where N* represents an antineutron. In this work we shall be primarily concerned with the antiproton, since the antineutron would be much more difficult to observe. Nearly all of the calculations made by meson theory of processes in which nucleons occur in intermediate states, such as meson production (cf. Section VII below), assume the Dirac theory and hence the possible existence of the antiproton. For instance, the observed decay of the neutral meson into two gamma rays is usually explained by saying that the neutral meson turns into a. virtual proton-antiproton pair, which annihilate by their coupling with the electromagnetic field into two photons. Th e antiproton was considered in certain attempts to explain the origin of cosmic radiation, such as that of o. Klein0, -5- who assumed the existence of whole galaxies of "reversed matter". In these, nuclei were composed of antiprotons and anti- neutrons, and surrounded by positrons to form atoms. If such a galaxy of reversed matter were to drift through one composed of ordinary matter, the nuclei would annihilate into energetic mesons or gamma-rays, and the mesons would ultimately decay into electrons and neutrinos. These electrons were supposed to constitute the cosmic radiation. Since it is now reasonably certain that the primary radiation consists of protons and heavier nuclei, and that the content of electrons and gammarays is negligible, such theories as that of Klein are no longer considered. On the other hand, N. Arley 4 assumed the primary radiation striking the atmosphere to consist of nearly equal numbers of protons and antiprotons. The antiprotons were to annihilate with protons of the air nuclei to form energetic gamma-rays, which in turn initiate the cascade showers observed as a part of the soft component of cosmic rays. In order to fit observa- tions of the soft component, annihilation cross-sections on the order of lo- 25 cm2 were needed. This could not be obtained from the Dirac theory, which yields values of less than lo-33cm2• At present, showers are believed to be initiated by the gamma.r ays produced in the decay of the neutral mesons, which are formed in the high- energy collisions between the primary particles and the air nuclei. One process by wh ich antiprotons might be produced in the atmosphere is the collision of an energetic pimeson with a nu- - 6- cleon; e.g., n..-. ., N -- P- + 2P, or rr-. . P ~ P- + P ... N. The first attempt to estimate the cross-section for such a reaction was made by McConne11 5 . He used the Weizs~cker- W illiams method (cf. Section VI below) in the following way. First he calcm- lated, using the Reitler theory of radiation damping, the crosssect ion for the reaction n+.., n-~ p- + P. Then in the colli- sion of an energetic meson with a nucleon, he transformed to a Lorentz system in which the meson was at rest, with the nucleon moving at high velocity. The field of the nucleon was represented by mesons, which reacted with the resting meson to make a nucleon-antinucleon pair according to the above process. Now, at the time, it was thought necessary to use the theory of radiation damping in order to avoid certain divergences and other anomalies in the meson theory, but since then it has been shown that this method is unnecessary and even gives unreasonable results in certain cases6. Besides that, the method is so difficult to apply that results can be obt a ined only in the limits of very low or very high energies. Hence we have calcu- lated the cross-section for a typical meson-nucleon process using the Feynman methods of perturbation theory (cf. Section IV). It is assumed that, despite the difficulties of the meson theory, this will give a rough description of the behavior of the cross-section at different energies, as well as an order-ofmagnitude estimate of its size. Antiprotons could also be produced in the upper atmosphere in the collisions of energetic primary cosmic-ray particles with nucleons of the air. We have attempted to estimate the -7- probability of such processes by using the Weizs~cker- W illiams method (Section VI). For collisions at extremely high energies, greater than about 101 2 e.v., it is known from studies using photographic emulsions that large-scale disruptions of the nucleus occur, with several mesons and nucleons thrown off. For such occui;::ences, it is best to use a statistical method such as that of Fermi 7 to estimate the comparative numbers of mesons and antinucleons formed. Once antiprotons are formed, what becomes of them? Clear- ly annihilation into two gamma-rays by collisions with a pro- ton would be an improbable event, for the same reasons as those noted above for the unlikelihood of production through the electromagnetic coupling. On the other hand, annihilation with either a proton or a neutron to produce mesons could occur. The probabilities of such events are calculated in Section III below. 1 ~ * The annihilation of antiprotons and_protons to produce mesons was first considered by McConnells, using the theory of radiation damping. Because of the unreliability of this method, the cross-sections have been recalculated using ordinary perturbation theory. Inde pendent work by Ashkin, Auerbach, and Marshak has already been publishedB. They give details concerning the distinctive appearance of annihilation events in photographic emulsions. -8I. KINEMATICS o:f THREE-PARTICLE PROCESSES. For later reference we wish to develop notations and for- mulas for the kinematical relations involved in processes in which three particles are produced. The treatment will be rela- tivistic except where otherwise noted, so that the term 11 energy of a particle" always refers to the total energy, including rest mass, and not to the kinetic energy alone. Masses and mo- menta are measured in energy units, being multiplied by c2 and c respectively, where c is the velocity of light. In the center-of-mass (c.m.) system, the energies of the three particles produced are E1 , E2, and E3 respectively, and the ordinary momentum vectors are Pl' p2, and ~0 . These are ' I\ " combined in the four-vectors pl, p/\2 , and p3, so that ( 1. 01) Then if T is the total energy available in the c.m. system, and if P" denotes the four-vector (T,O), we have (1.02) by the usual conservation laws of energy and momentum. We introduce the scalar product of two four-vectors as follows: ( 1. 03) where pl, p 2 are the magnitudes of PJ_, p2 , and B is the aD.gle between these three-vectors. Then if m1 , m2 , m are the masses 0 of these three particles, (1. 04) - 9- Thus we can use (l.02) to obtain ( 1. 05) (P"' - P1 I\ 2 "' 2 P1 "'" P2 " 2 -:. ill3 2 : P3 I\ p 2 ~ A p.-,) 2 -:::; ~ I\ ,,.. 2P" • p ~ "' A 2P • P1 2 1' ,.. i::>pl • P2 : .,. m2 2 - 2TE1 - 2TE 2 + 2(E E2 - P1P2COS 6 ) T2 + m12 1 or ( 1. 06) where ( 1. 07) Thus for a given total energy T, and a fixed momentum 'Pi for particle 1, the end of the momentum vector p2 lies on a surface of revolution about Pl given by (1.06). We shall need the extreme v alues of E2, P2, which occur for fl -=. 0 and g = 11 • Let ( 1. 08) E2 = E2'' P2 = P2' for (} -=- 0 ( 1. 08) E2 "" E2" ' P2 = P2" for 8 = Tr. From (1.06) we see that these are the roots of the equation (1.10) P12 P22-=. Pl2 (E22 - m22):. ( tF2 - (T - E1)E2 ] 2 or ( 1. 11) Then we must have ( 1.12) E 2 11 + E 2 1 -::. F 2 G- 2 (T - E1) 1 ( 1.13) E2" - E2' : P1G-2 [ G2 -(mz + m3) 2 ] 2 ( G2 - ( m2 - m3) 2 ] From these it can be shown that (;L_.14) l 2 -10- (1.15) Now p 2 11 + p 2 1 is the longest diameter of the p2-surface. As the energy E1 increases, this diameter decreases, and it vanishes for the maximum value E~. a2 - (ma ~ m3)2 0 s o that This occurs when = E~ = [ T2 ~ m1 2 -(m 2 ~ m3 ) 2 ] /2T (1.16) VV hen E 1 0 = El' (1.17 ) E3' = E3 " = m2m$ m3 ( T 2 -m12 ; (mz + m3)2 J In this case the ~ 2 -surface has shrunk to a point, and both p2 and p3 are in the direction opposite to p . The other extreme 1 is for E1 m1 , Pl 0, in which case symmetry shows the p 2 - = = surface to be a sphere, with F2 (1.18) Ez = 2(T - m1) = We no w wish to derive an expression for the differential volume of momentum space (1.19 ) dT - r· E,J E, dn, r2. E~ J E, d x Jo{ d 3 f1 dfl rz. dT dT where dJ2, is an element of solid angle in the space of Pl' <and ~ is a polar angle which, with 0 , locates the position of the vector -P2· x -=.. cos () . Using ( 1. 06) ' dx ~ dT ( ~~) . E2. ~ E0 P1P2 dT so that ( 1. 20) runs from Oto 2 n , wh:ile E2 1 ~ E2 ~ E2 11 • The integration over E1 is carried out last. The factor In integrati ons, cl E1E2E3 will always be cancelled out by other terms, and it is -11- clear that any pair of particle momenta can be used in d T . We now calculate the total volume of momentum space available to the three particles produced in the reaction, in the non-relativistic limit where ( 1. 21) and none of the masses van ishes . E1:. m1 , etc., in (1.17). that over E2 gives E2 11 The integration over E2 1 , - In this case we can set ol gives 2 '1f , and that over 11, gives 4 rr , so that, using (1.13), (1.22) Jcl-r = 8n ~3 f E;, ~ ~· ( G'-- {..., + ""• J') i [ G, - ( 2 · l'VI, W'2. ""> - .,., ) 'J i M1 In all these terms except the first bracket, we can assume ~ E 1 -- Eo 1 - ml' "'o In the first bracket, how- ever, we put, by (1.07), (1.23) G2 -(m 2 + m3 ) 2 : T2 - 2TE1 + m1 2 (m2 + m3) 2 = U(T - m1 + m2 + m3) - 2T (E1 - m1) Non-relativistically, (1.24) 2 G - E1 - m1 ( m2 + m"<:) 2 :. v = p1 2/2m1 so that 2U ( m,, + nh) v &;, m 2 .:.El.._ m1 -- where p 0 is the maximum value attained by p 1 , approximately. We shall need also the volume of available momentum space -12- in the case of four particles of equal mass M produced in a reacti on with total energy T in the c.m. system. this we use a method due to Fermi 7 . To calculate Again let U : T - 4M be the excess energy, i.e., the total kinetic energy of the final Then, non-relativistically, particles. ( 1. 26) P12 ... P2 2 ... P3 2 + P4 2 = 2MU ( 1.27) Pl + '.P2 ... P0 ... P4 - 0 - Introduce the new vectors q, r, s- given by Pz :. q - r - -s - - P4 = - q - r + s = q ... r t s, 'P0 -=- q + r - s, ( 1. 28) p 1 Then q2 + r2 + 82 :. -!MU ( 1. 29) where (1.30) where .., q~ = qx 2 ... qy 2 + qz 2) etc. It is seen that d.3 !>1 d 3 P2 d 3 "P3 - J.3 d.3 q_ d 3 r ct3 s - J ~ - 1 1 1 1 -1 -1 1 -1 ~1 tr ansformation (1.28). ( 1.31) - 4 is the Jacobian of the Thus fd-c = f .!' f, ~'+, d f3 3 where the latter integral is the volume of the nine-dimensional sphere (1.29), the radius of which is R ~ .../ -!ViU. Using the well-known formula for the volume of an n-dimensional sphere, ( 1. 32) 2 n -13- the integral becomes ( 1. 3.3) -14- I I. AP PL I CAT I ON OF THE FEYNMAN METHOD In these calculations we shall use the methods of R.P. Feynman9 , and our notation will be the same, except where otherwise noted. In particular we use his convention as to the Dirac y,... , with the definitions r'f = (3 y, .::::. ~ ~)( etc. > y matrices (2.01) I J f3 ~. We use the notation (2.02) Yr Pr ::::: p :::: ~ [ F" - oe. r) , Considering only the charged mes on theory at present, we state the following prescri ption whereby the matrix element of the collision operator H in momentum space is written down fror.1 the Feynman diagram for the gi ven p rocess: For each virtual- nucleon line there is a term (~ - M)-1 where a is the four-mo- mentum of the nucleon. For each virtual-meson line we put (q 2 - p2) -l where q is the four-momentum of the mes on and is the mass of the meson. p At each intersection where a virtu- al meson is emitted or absorbed, there is an interaction term of the form where g is the meson coupling constant, and f (p,) and t l Ff) are four-component plane-wave Dirac wave-functions for the initial and final nucleons.* 0 is an operator depending upon the The factor of f 4-Tr alized units. occurs because we are using unration- % -15- type of coupling. For scalar coupling it is 1, for pseudo- /"s- , and for pseudovector Ys '[ /~ · \V lr) = 'f* (r ) (3 is the adjoint of f lp ). At an intersection involving a real scalar meson of energy E, the interaction term above is multiplied 1 by (2E)- 2 . In determining the cross-section for the given process, the matrix element thus written down will be multiplied by its adjoint and summed over the various spin states of the initial and final nucleons by the spur technique presented by Feynman 9o As is there pointed out, it is necessary to multiply by a facfor each real nucleon involved, in order correctly to adjust the normalization, where M is the mass and E the energy of the nucleon. The differential cross-section for a process resulting from the collision of two particles of relative velocity vis then 2rr ( 2. 0.3) where -kv 2 H\ av 2. I ~I \ o.v f F is the absolute square of the matrix element of H summed' over the spins of the final particles and averaged over the spins of the initial particles. fF is the usual densi- ty of final states in momentum space. As an illustration of the above prescription, we shall work out the cross-section for the annihilation of a proton and an antiproton into two charged mesons. In the c.m. system the proton is represented by the four-vector p1 : (E, p), while the antiproton has (E, -p). The process, represented by the -16- Pl r P(r~) 1) (Fig. 2.1) diagram in Fig. 2.1, consists of the proton emitting two mesons qI\ = (E, k -) ,.. and q' state p2 ~ (-E, = (E, -K) and g oing into the negative-energy .,-:. p). The intermediate state is a neutron of momentum (2.04) - - " ,.. " /\ " PN :. ( 0' p - k) ::. pl - q ::. P2 + q ' Then the pertinent matrix element is (2.os) H = 2 1.. ~ ' - rr1E '-- c_ t {Fi.)0 2 (p"'-M) -I 0, tlr, ) The relative velocity of the initial nucleons is (2o06) v = (2p/E)c The density of final states is d3 k (2.07) k E JJl where T -= 2E is the total energy, and d D ::. sin (} d B d 'f the element of solid-angle for the meson direction. additional factor of is We get an t from the average over the spins of the initial nucleons, so that the differential cross-section is (2.08) k E d .fl =~ (.L) _I L 11-112. 1'c 2p Lf 2 { 2 n l c. )3 where L indicates that \Hl2 is summed over all possible -17- f (p 1 ) satisfies the Dirac equa- initial and final states. tion (2.09) In the case of pseudoscalar coupling, = 0 'Y~ , and we can simplify H by writing (2.10) ~ lyi.) ys- lfN -M )-\ Ys- t lf') - t (r,) ys- l rN + fV\) Ys- ~ lr· ) l f",_ - M ~) _, - !{r2.HfN-""') tlr, )== ifilr4Hf,-Cf-M)t(r.> _ ~lpL)~+lr,> ,...,_ ML.. )-i. PN -l f-k - M f-"- +M '2._ ( 2 .11) . H = E 2[~ '1.. - ( - "L. • )'- t. 1.... ~ P- J.. ) M~ J H r, J ~ o} ( r .,c.+ I ) Taking the absolute square and summing, L \ 'fl r~) 1 2 12 ( • ) *(p '2. ~ t Mf sr (r 2 I = M - L [ 2 (fl-· 1) (f 1 l FI M ) 1 2-f M) 2 ) i ) + i fl FI -P t 2 I . ( 2. ) l. ] -2 k"2.. (E~-p'\._cos._f1-) M-2.. This must be multiplied by -M 2 /E 2 , as noted above, so that ( 2. 13 ) L I l-11 ~ -:::: '\..j__ 1. ( 2rrq ~ c '"l.. ) '2.. · 2 I1c. '- ( t- '1.. -p' - cos~ f'I) t1 .J E. .. [ lf-k )'- + M'l-) i. . Hence the differential cross-section is (2.14) j~ k. 1 (E'--r .. co5'"~) dfl Br E' [ ( r-k\'2. + M'-]2. in the c.m. system, with (p - k) 2 ~ M2 :: E2 "t k 2 - 2pk cos e :. 2E(E - p c os e )- r- 2 It is of interest to derive our formula (2.05) from the -18- perturbation theory of Feynman in this simple case. We note that we desire the transition amplitude A(2,l) from an initial plane-wave state (2.15) f{l) =- l2n r 3/ tfr,) e-'r··1<, ) f• = lE,, f•) A 2 A to a final plane-wave state :lJ "' /\ f' =- (-E2.,-f,) j (2) = (2n)- i.t(f2) e-c.·p2·>'2 (2.16) - where E1 , E are the energies and p1 , P2 the momenta of the 2 initial proton and antiproton respectively. This transition amplitude is given by (2.17) A(2J1) =:-JI 5 (1)~ k'o.J(2,\) ~+ (1 )d3x,J3 ~ where ( 2 • 18 ) kt 2) ( 2 . \ ) -:: Here K+l -JJK. l 2 , v(q ) k'+ l 4 3 ) vl '<. LI ) 2. h J 3) I( 3 I )d T3 d T'I I is the free-particle kernel for the initial proton, given by ( 2 • 19) l<+ 1 l 2, l ) ;:::;; I i. n) ._. 2 J ( f• - M ) -I e 'L p( - i " ,... )d4 p1 F 1 • lC 2 1 = where ~21 : i2 - ~l ~ (t 2 - t1, x2 - x1) and d 4p 1 dE 1 ct3p1 . ( We take t =- c :. 1 in this discussion.) Similar expressions h o ld·. f o r K.f.n a n d K d .,..« -- sions and using J +2 ( 2 • 20 ) v f { 3 ) ;::: dt « d&.x . 3 v comb.in1ng . th ese expres- I(+ 1 (3 , I ) ~ {( I ) J l ii 1 with a simi l ar expression for g(4), we find Al2,l) (2.21) =- tl2lT)-l.f ff jlLf) Vt~) (p~-M)-'. e>tr(-ijN· Xn) V(s) {l 3) dT3 dT'I d" FN. -19- Now at the space-time point 3, the potential acting is given by charged-meson the ory to be V(~) -=j ~ 0 1 (2.22) YO: ,. where q1 = (Q1 , Ci.1 ) , q2 = ( Q.2 , q_ 2 ) are the four-momenta of the emitted mesons. '"t"Pw 0 is the coupling operator used above, and is an operator wh ich changes a proton into a neutron. Similarly, (2.23) Since (2.24) eral reference system. ~ Now integrating over the space parts A A of x 3 and x4 , and then over the space part of PN' Al 2 (2.25) j \ ) = i L2" ) - • j ff J r~ d i J t "' ~ ( 'f, - f f"~ )· 't Cfb i. l 2 n ) - I __ l,,-r . +r~ ) e 1 i[Q2.-E2..-E..,)!., e 1 - i(Q 1 -E,+E,.,)i:~ ff f d EN d +~ d ·L+ ~ l ~, + ~~ - p. - pi ) · e dQl..-El..-E...,)-1:-,. e i(Q,-E,+E,.,)t3 H J-1 . Let E1 + E2 : E1 : initial energy, Q1 + Q.2 ~ Ef: final energy; P1 + 'P2 = Pt: initial momentum, q 1 ~ Ct2 ~ Pf: final momentum. H can be written ( IE.., I :::. fr"' l.-t- f'l\'1. ) j -20- (2. 26) 2 "L =- ~ 1-\ 'JQ, ai. f (r > o ~ ( E N yq - :y ·1s,,, + M ) o . '"" t r, > (EN - I EN I ) (EN + IE"' I ) 1- Now in integrating over EN, we take S tesimal negative imaginary part, ) EN I 0v to have an infini- The integrand contains Thus for t 3 < t 4 we can complete the contour in the lower half of the EN-plane and evaluate the integral by taking the residue at EN~ + V PN 2 ~ M2-iS = fEN I - i o , while for t 3 > t 4 we must c omplete the cont our in the upper half of the plane and take the residue at - \EN I + i ~ . ( = a E : N Thus we get for t.'3 < t4' using the 0) . Cauchy residue formula, l -2~ (2.27) f d- e t ~ -·E l (-l "' '+ -t) 3 ~ (r1. ) 0 L ( IE~' y., H = --t e -ilE...,ILi'(-tl) . z -rw. y + M ) 0 'P (EI ) \E,..,l I -21- where the integrals over t , t 4 are extended over the time of 3 observ a tion of the system, 0 to T. Carry ing out the integration on t 3 , we get A{2,1) ( 2 • 00 ) e f o T d ·l-..1 2. =~ - 'IEI };lr+-ri) · YQIQl.. 2 "' f e l. ( EI' - E . ) t u .. , ,- e d Q. - E. - IE,..,J H.., - .. i(Q 1 -E,-1-IE"'I) [ q:- { r"2. l o 2 ( \ E w 1y., - i. ( QI - EI - \ E tJ ' ) T e i ( Ql. - [ ~ l Fi.) t> 2. ( - I ErJ l y., r,.,. y + M ) o, +(r, )J + E i.. + IE") } -\: L4 t' ( QI - • -e E I - I E rJ ) ) - f w ..y + M ) o, i l Ef - E" l ) i., * (r I ) ] ) Now considering ti mes T long compared to ~ over the v a rious energies involved, only the terms in exp i(Ef - Ei)t 4 will contribute to the integral over t 4 , the others averaging out Note that the terms in the brace are just those one would obtain in writing down the matrix element for such a process using the methods of the Dirac hole theory. They can be recan- bined to give (2.02) The A(2, 1) H S' ( pf - P·. l. ) b -function merely expresses conservation of momentum in the process, while the exponential term g ives conservation of energy for long times of observation T. Thi s expression {.32) -2 2- is then handled in the usual way to obtain the cross-section f ormula (2.03), with H given by (2.05), replacing Eby YQ1 Q2 • -23III. ANNIHILATION OF ANTIPROTONS. (a) , Annihilation into Charged Mesons. As shown in the previous section, the differential crosssection for the process (.3.01) with the pseudoscalar coupling is, by (2.14), 3 2 d CS" = 2.-n-5'f1 ( E -p2 co5~eJ s.·Yle de 8 p £2. ( E' + 7· - Zf' cos ())2. 1 (.3.02) To get the total cross-section, we integrate over the angle 9 , so that ( .3. 0.3) t5 Let E0 , p 0 be the energy and momentum of the incident antiproton in the system in which the proton is initially at rest (l aboratory system). (3.04) 1 E2 :. 2M(E 0 Then 1 E + p_ + M) , p 2 :: 2M(E - M), 0 E - p Eo + Po --M- as the total cross-section for annihilation in the laboratory system. Non-relativistically this is / :.::: - , 0 "' rrr 2. -c 0 [ v where r 0 :: g2/M:. 2.1 x io- 14(g2/flc) cm., and vis the relative -24- velocity of proton and antiproton. (3.07) _, 0 _:_ ~' 4 rr r D'2.. - For very high energies, .t1. ( I oq -2. Eo - f ) EC> J The scalar theory gives, with 0 which becomes, up on integration, 6'-::: Trj'li [ BE~ I + (3.09) 4H"-1-2E"--r'-10 r - zf'l M · = 1, M2.+(r-•1f--_(4H"l.-r°1.t]. j M"ltrr-;r- r"+Lf-M"l.1' This reduces to (3.06) and (3.07) in the limits of low and high energies, respectively, so that we need not consider the scalar case further. Another interesting case is that of pseudoscalar mesons I ,.... with gradient coupling. Here we have 0 ::. rs~ comes I , so that H be/ (.3.10) since (3.11) (3.12) t (pl) : -(PN - M) t (pl) 'f (p2)Ci2 = t (pz) (P°N - P'2) = t (pz) (pN - M) . ql o/ (pl) = (pl - PN) We can write (3.10) as Squaring and averaging by the usual spur technique give -25- (3.14) Z M'l. r Lf e o S f) If we integrate this expression over 2 order p2 /M , we find at low energies (3.15) ~ . :. . 9 and negl ect terms of 1rj.., M'- ~ ,,.. ., v ' At very high energies, on the other hand, we get (.3.16) d ..:. rr :J" M E0 {, f4 'f. Thus the cross-section for annihilation increases with the energy E0 of the incident antiproton. It was this anomaly which 5 led McConne11 to ap ply the Hei tler theory of radiat.ion damping to this calculation . But as we have seen, the p seudoscalar di - rect coupling yields the expression (.3 . 07), which drops off at high energies in a reasonable manner. When McConnell wrote his papers, the pseudoscalar direct coupling was i n disrepute because of a misconception as to the nature of the nuclear potential it yields. This difficulty has since then been cleared up. (b) Annihilation into Neutral Mesons. In the following work, wherever the effects of neutral mesons are included, we use the form of me son theory which assumes coupling through the isotopic-spin operator ~l , so that there is a difference in sign de pending upon whether the neutra l is emitted by a proton or by a neutron. mes~ For a virtual neutral -26- meson, the sign of the term is positive when emission arm atisorption are by the same type of nucleon, and it is negative when the meson is emitted by one type and absorbed by the other type of nucleon. The coupling constant will be g 0 • For the symmetric theory of Kemmer, take g 0 -::. g/ -{2. We shall now study the following process: pl- (p) + p- (3.17 ) (- p) - no (q) + no (-q). In this case there are two possible Feynman diagrams, depending upon which of the neutral mesons is emitted first. These are given in Fig. 3.1. \ no(il.) \ \ \ \ I\ " \ s i p lf,) P cpJ..l (Fig. 3.1) (.3.18) A A where r, s are the four-momenta of the intermediate protons. Then the operator H, in the pseudoscalar theory, becomes H = 211jol.E- 1 [ \F(ri.)ys (r-M)-'y.r t(r,J + (3.19) since f f;l.) y-; ( s-M )-' ys +rr,)] = - 27- This gives, in the center-of-mass system, { 3. 20 )J<> 4 3 = jo j.. p cos'e [(E.'l..fj'l..t-4f'l..j'l.c.os'-9yl.{J s:viScledr 2E 2 ( ( E'l..+ 'L.J1. -4f"l.'t1. cos._ 9]2. 1 The integration on <p runs only from 0 to 'ft , since the two neutral mesons are indistinguishable. e, (3.21) Integ rating on f and ~ == rr:y .. 'i '!- } - I + l 4E~ I 4pi ( 1 L. 'L. E +1 -I- t: 'I )( 'l.. l... M +(p-t j) _ Z(E'tfJ oj M'lt (r-1'... µ. L J . 2(,'i +4M\tJ Transforming to the laboratory system by use of (3 . 04), and Non-relativistically, with v the rel a tive velocity of proton and antiproton, ( 3. 2.3) while for very ( 3. 24) Thus we s e e that annihilation at low energies is predominantly into charged mesons. This also follo ws from the fact that for very low energies, the initial nucleons form an S-state with J : 0 or J : 1. The par i ty of such a state with a particle and an antiparticle is odd. The neutr a l mesons, having spin zero and Bose statistics, can be produced only in st a tes of even parity and even J. Thus the annihilation into neutral me- sons is forbidden in the limit of zero relative velocity, be- - 28- cause of the necessity for the conservation of angular momentwn a nd parity. (c) (This argument is due to Mr. M. Ruderman.) Annihilation with a Neutron. Since our theory treats protons and neutrons on an equal basis, we must expect that an antiproton could react with a neutron, producing a negative mes on and a neutral mes on: (3.25) Here again there are two possible processes, . illustrated by the accompanying diagrams (Fig. 3.2). n-( ;.> / '\n°(i2) I \ \ \ I \ \n-l\. J \ no(f1) / \ I \ I \ ( PJ p ( f1.) I \ I I " r \ In the former case, the I I I s" (N) N{f.) (Fig. 3.2) intermediate nucleon is a proton; in the latter, a neutron. As noted above, the symmetric theory then requires a difference in sign of the corresponding terms of H, which we can write down as follows: H == 2 IT J j <3. 26) 2n J J E - I [ ( 0 E- I *l 0 f t ( f y r - 1- \ 'Ys- t l r 2. ) S" ( M I ) r , ) 1 :::. r. .- M ~ )-' + ( s~ - M ~ ) - \Jf l r2. } i + (r Pi. ) Y~ c s- - M ) - I Ys- ~ ( I I ). This gives the differential cross-section in the center- ofmass system: -29- (3.27) In the laboratory system this is, to terms of order p4 /M 4 , (3.29) tS = ~~ nqL..cioL. [ 4 . I - JtEo M)J E M M I j '- -t - E 0 + Eo O 0 M At low velocities, (3.30) . fr:J .,_ jo 2. 6 = c v M .._ ( I- [~~) while at high energies, (3.31) <f · lT J 2.. jo '2... Z.Mi... Thus the cross-section for annihilation of an antiproton with a neutr on is twice that for annihilation with a prot on, nonrelati vistically, if we put g 0 2 : tg2 as in the Kemmer theory. (d) Annihilation with a Bound Neutron. It is of interest to consider the possibility that a slow- ly moving antipr oton will annihilate with a neutron bound in a nucleus, with only one meson being emitted, the nucleus taking up the extra momentum. In this case, we can write the wave- function of the neutron as a unit Dirac spinor for a state of positive energy times the Schr"dinger wave-function f (r) of the neutron, supposed bound in some sort of potential well - 30- which represents the rest of the nucleus. The neutron will e- mit a negative meson and become a proton of negative energy, filling the hole which corresponds to the antiproton. Thus the final wave-function will be a unit Dirac spinor for a state of negative energy (and proper spin) times a factor exp [ -ip' · 'f/r1c] , where ~· is the momentum of the antiproton. With pseudoscalar coupling, the Hamiltonian operator wi l l be (3.32) H = _,j'r..cift JJ'r ( +/ ~Ys-+.,) e-<'f·"/t.c where q is the momentum of the em i tted meson. Err ergy of emitted meson ; 2M. t ions, combined with ~ Ys- =-v,;·'-+ f"' = en1. The spinor parts of the wave-fu newill give unity, provided the nw.- tron is of spin opposite to that of the antiproton, and they give zero otherwise. Thus annihilation will oo cur with about half of the neutrons in the nucleus, namely those of proper spin. Thus (3.32) becomes ( 0. 3.3) H =- j n c. y-E,.; . l.. fE l jd 3- "' ( - r T r ) e .: l f, -1). r the integration being taken over all space. order of 2M, while /t\. c.. Now j ~ f is of the l"P' ) is small compared to M, in the non-rela- tivistic limit we are considering. Hence (3.33) represents the Fourier component of the bound-neutron wave-function corresponding to a momentum of about 2 Bev. (3.34) H - -ij where we have taken ~ ( .3. 35) (2rr) Let ~ : -~' + q. 4 YE-rr = c = 1. Also Then -31- is the momentum representation of the wave-function of the bound neutron. The cross-section for annihilation is then (3.36) ($' ::= - 2rr i:. v -I 2.. 1''12. M f- t- vvhere v is the velocity of the incident antiproton, and fF is the density of final states, given by (3.37) (3.38) J 4M 2 - f 2 :'. 2M. To compute f (p), we write the Schr~dinger equation ~~ 2 ( .0.09 ) - - 'Vt +Vt ==Et with q-;_ 2M where V is the potential binding the neutron. Now substitute the inverse of (3.35): (3.40) so that (3.41) (2nfij(E-~) cy(p )e'f·" J';; =V(r)t(r). Take the inverse Fourier transform of this equation: Now for \'P \ :: 2M, the exponential varies rapidly, while t (:r), being the ground-state wave-function, varies relatively slowly over the nucleus, where V(r) is different from zero. can replace + (r) by its average ovar the nucleus, write, since p 2 /2M >> E, Hence we t"- , and -32- l ==-2Mp-i.l2n)-i+~ v(r) where v(p) is the Fourier component of the potential for large momentum transfer p. This can be estimated from high-energy neutron-proton scattering, for the differential cross-section of which the Born approximation gives the formula tv'\ "l. l. ( .3. 44) d 6' sc =l.. I v JI d tr 411 Replacing d.Q by 4 rr section, <15 , , fl . to estimate the total scattering cross- we can combine (0.38), (3.4.3-.44) to obtain ( .3. 45) We can esti:nate j ~a.. I'- by the reciprocal of the nuclear volume per nucleon, which is approximately lt~I~,_ a- 3 , where a=:' 1.5 x lo- 13 cm., so that 2 1+~( M- 3 ,-...,1 (M8 ) - 3 ; .Ma:: 0.14. ( .3. 46) At the highest energy available, about 270 Mev, the total n-p scattering cross-section 10 is ~c..; 0.038 x io- 24 cm 2 = 86(11c;iif. This corresponds to a momentum of about 0.38M in the center-of1 mass system, or an average momentum transfer of about ~M, so that our result will be somewhat too high. We get for this one-meson annihilation cross-section (3.47) lT 2 )' v ( 0.2) M ~ (~c. ~c. c. This is seen to be a bout 1/5 or less of the cross-section (3 . .'.D) for annihilation into two mesons. (e) Annihilation in Matter. When an antiproton passes through matter, it may annihi- late by any one of the processes (.3.01), (.3.17), and (.3.25). For energies of more than 1 Bev. or so, the annihilation crosssections are small, and the antiproton may be expected to lose energy by the same processes as does an ordinary proton. The ionization energy-loss would be approximately the same for an antiproton as for a proton at all energies. For kinetic en- ergies less than 1 Bev. or so, the cross-sections d 1 for annihilation with a proton into two charged mesons, and <S'l. for an- nihilation with a neutron as in (.3.25), will become large, and the antiprotons will disappear in this manner. We can neglect the cross-section (.3.22) for annihilation into two neutral mesons, for this is smaller than the others by a factor of nearly 1/100 at these low energies. Let n be the number of atoms per cm3 ., each atom contain.,_ ing Z protons and N neutrons. Then the probability that the antiproton is annihilated in going a distance dx is ( .3. 48) are given in (3.05) and (3.29) respectively. In a distance dx, the antiproton loses an energy dE =(dE/dx)dx. Thus in slowing down from an energy E2 to energy E1 , the probability of annihilation is ( 3. 4 9) y ( E.. , E • ) == "' I z E,_ ( E, 6", ... N 6".. ) ( ~: r dE -34Values of the energy loss (dE/dx) and the range of protons in S.T.P. air and in aluminwn hawe been calculated by 11 Smith • We use his results and evaluate (3.49) by numerical integration, taking g 0 = g/ {2 and g 2 /fic = 1. (For ant ipro- tons in air of kinetic energies less than 10 Mev, we use the energy-loss curve for protons given by Bethe 12 .) Thus we obtain the following tables, where E 1 , E 2 are kinetic energies in Mev, and h R is the range traversed by the antiproton ~ slowing down from energy E2 to energy E1 • Air E2 (Mev) E1 (Mev) 2000 1000 400 300 200 100 45 25 1000 400 300 200 100 45 25 15 10 6 6 2 2 1 0.6 1 0.6 0.2 y 6 R (cm) 5 1.89 x io- 1 4 • .36 x 10 1.45 2.23 2.58 x lo-2 2.82 x 10 4 2.54 2.34 2.37 1.67 5.40 x 10 3 1.09 0 3.2.3 x io1.11 1.22 0.352 4.05 x io-4 68 5.42 40 1.29 4.85 1.02 1.30 0.88 0.76 {). R (~~(}) 564 289 .36.5 30.3 21.6 6.99 1.44 0.455 88 x lo-3 52 6.3 1. 7 1.0 -.35Aluminum·* E2 (Mev) E1 (Mev) 2000 1000 400 300 200 100 45 25 12 1000 400 .300 200 100 45 25 12 y 6 R (cm) 9.9.3 x io- 2 7.69 1.38 1. .36 1. 27 5.92 x 10- 0 1.77 9.56 x lo-4 5.11 .3 218 112 14.2 11.8 8.51 2.77 0.574 0.226 0.077 It is seen that an antiproton of a few Bev. energy has a good chance of annihilating in flight, with a probability of about 0 • .35 (g2/1'ic)2 for a kinetic energy of 2 Bev. in air .. If it escapes annihilation in flight, the antiproton will be slowed down and eventually stop, being captured by a positively charged nucleus. Wightman 14 has calculated the modera- tion times of antiprotons in hydrogen gas and finds times on the order of 10-lO sec. for slovdng down from 10 Mev. to capture by an H2 molecule. In heavy substances the times would be expected to be somewhat shorter than this. We note that the antiproton would arrive in the lowest Bohr orbit about one of the nuclei by radiating photons in a time TR given roughly by the ordinary radiation formula 2 (.3.52) * 't"R -I ,......,, e v 3 ~ 2.. t.. c .3 Note that a standard photographic emulsion such as is used in nuclear investigations has i stopping power and average Z comparable to that of aluminum 3 . Hence we can use these results for aluminum to get the annihilation probability in an emulsion by multiplying the Y of the table by the ratio of the ~densities, which is about 1.45. where a is the Bohr radius fi2/zMe 2 , while h v would be on the order of the ground-state energy, (ze 2 /11.c) 2Mc 2 . Thus we get (i: )s ~ or 'lt\ - 1 ,_, Z Lf The lifetime TR _, z- 4 . 3 • 10- 14 sec. 't'o.. of an antiproton in the ground state a- bout a nucleus can be estimated by noting that it is given approximately by the cross- section Z 6' 1 + ..1v1 6 2. ,..,Tr'-( r., t. +2 N )cv 4 multiplied by the relative velocity v, times the density l'f(o)f2. = of antiprotons at the nucleus. origin of Here ~ (0) is the value at the t (r), the wave-function of the antiproton in the lowest Bohr orbit. Taking this as the hydrogen-like wave-func- ti on t l r) (3.53) where ~ L. c ~o (3.55) \\/\ I M' ei. l~lo)l T~ ( l:. rr rA 3 '2- we find (3.54) f -\ 2 I 0 )I e _Zr/a.. A = z + rv AM A+I ( ~e; f (~Jl rr __I_ 0 (A_, + l )-3 z+ 2rv ( t~r M ( t~)2 ~I -t A-') 3 k 8 - I ( We have divided by two to a ccount for the fact that the antiproton will annihilate only with nucleons of opposit e spin, in the non-relativistic limit.) This gives (3.56) Of course for most nuclei, the ground-state radius is of the same order of magnitude as the nuclear radius, so that the -07- antiproton will spend most of its time within the nucleus, and the annihilation time T0.. will be much shorter than the estimate ( 0. 56). (For the special case of the antiproton bound to a proton, see Appendix I.) This time is very short compared with the time taken by the antiproton to slow down and be captured by the atomic nucleus. -.38IV. PRODUCTION OF ANTIPROTONS IN MESON-NUCLEON COLLISIONS. The first production process we shall consider will be the result of the collision of a positive meson and a neutron: (4.01) This is described by the diagram of Fig. 4.1. p ( p3) _\. ___ ------------( "I - - "I " N l lco) (Fig. 4.1) Here we have, in the center-of-mass system, (4.02) (4.0.3) "k 0 = (E 0 , k) , k1 = ( E- , -k) P1 =(E1, 1\), P2 = (E2, P2), P3 = (E3, P3) The antiproton is considered as a proton going backward in time, and the associated spinor "' (p 1 ) satisfies (4.o4) c-P-1 - 11) t (p 1 ) = o For the other protons, (4.05) The intermediate neutron has the four-momentum (4.06) " 1 = P3 " - "k1: Pl+ ,.. "q' P where ~· is the four-momentum of the intermediate meson: (4.07) f\ q' = ko - P2 = P' - Pl A A A ,... Since the emergent protons are identical particles, we -39- must consider also the pr ocess obtained by interchanging ~2 f\ and p 3 in the diagram. The momenta of the intermediate parti- cles are then designated by a double prime: = p 2 - k l' I\ (4.08) " P ll /\ " I\ " q" ::. ko - P3 We note that if T is the total energy in the c.m. system, (4.09) 2TE 0 = T2 + M2 - f2, 2T e :. T2 - M2 + f 2 .> T : E 0 +e . Dividing by two to average over spins of the incident neutron, the differential cross-section for the process (4.01) is given by 2 d 6' = -' lf ' \ ' I f-11'2 p 2.. ~v L. I F (4.10 ) The relative velocity of neutron and meson is v, given by k k kT -vc -- -Eb +c .....-sE0 (4.11) The density of final states fF is d 3 f1 d 3 f2... (4.12 ) (2rrt..c)&dT where d i:- is given by (1.20). L, \Hf2. is the sum over the states of the initial and final nucleons of the proper transition matrix element in momentum space. Assuming pseudoscalar charged mesons, this can be written down from the diagram of Fig. 4.1, according to our prescription of Section II. (4.13) H We thus obtain (4rr) /a. 'J 3 ~ 3 c.. 3 (2..c:)-i. z.-i (o 1 3 oi..) where (4.14) o, t(r2.)y~t(k 0 J (f'~r1-J-' · f (p11 y~ Cp-'-M)-'yr t(p,) and o2 is obtained by exchanging .P2 and p3 everywhere, changing p~imes to double primes. Requirements of anti-symmetry between -401 the final proton states cause us to write 2-~(01 place of 01 alone. o2 ) in Using (4.05) and (4.06), we can simplify (4.14) to obtain 0 (4.15) _ _ t tr~ J ys- ± u( J f l r ~ )k, t' (e, J o I (cf. (2.10)). (f''-M,.)(,,2_fA-i.) Combining (4.10-. 12), (4.16) Using the spur technique, and noting (4.04), we have t ( 4.11) L lo, I'= Sr lf z+ t1) (J;.+ M) 'Ys: 5r (f, + M) k, (-f,+M)°k, 4 l 2 M ).. p'.,,-M"l. ) 2 ( ,,~ -r ,_ ) (4.18) Sr lfi. t M )ys l~o + M )ys = - 4tML.- pz.-Ct>) = - 2 i' 2 ... ,,_ s r ( M + r3 ) k. ( M - f1 ) le. I ::: ~ f ( M~k,2- -f->~ k, f k,] :::- 2 r 4 (,; . t ) ( p. . k, ) - r-, (r-... r3 ) J ( 4 .19) 1 2. ( 4. 20 ) ( " ,.. ,.. L:"lo,\'=- i'1.f 4r1·k,J(p,·k, ) 4 M 'f ( ~ .~ -r- L) 4 { f 4 21 ( • -r (r,fr J "1. " .... ) 2. 3 f"l. - M "1.) l. b 0 -=- t t.e~1ys *l ~o *l ko )ys Y'(r,) ~ l ri.) k, tlf') ffp,J k, f(f1J J I 2 l''l.-r•-y t , .."1._r_ _ ) lf'~-M'L) (f"L-M'-) -41- Then we can write the normalization. It is easiest to complete the evaluation of the total cross-section in the case in which the total energy T is just above the threshold energy 3M. are non-relativistic. Then the produced particles We then have (4 .25) Us ing (1.25) we have (4.27) We note that if Err is the energy of the incident Tr -meson in the laboratory system (neutron at rest), and if 6E" is the energy in excess of the threshold energy of 4M - f2/2M, (4.28) U ~ ~ 6Eir ==1 (Err -4M + t-4-i/iM) Combining these results and substituting into (4.16), where (4.30) w _ r '~ti-,r'r1-,r~l- rt -:f~.t- -,_ ~~,. -42( 4 • .31) For energies much larger than the threshold, it is necessary to take into account the angular dependence of the expressions (4.20) and (4.24), and these must be integrated over the momentum space of the final particles. This integration is feasible only if cert ain approximations are made. of these is to set the mass of the meson p : O. duced by this will be on the order of p2/M2 . The first The error in- The second will be to neglect the first term in (4.24), which is an exchange term proportional to (pz - p0 ) 2 , and which vanishes in the nonrelativistic limit. (4.20) now becomes, after some simplification, using (4.06) and (4.07), - M ( 4 • .32 ) 4 ~ I o, I '2. i.J - while (4.24) can be written (4.0.3) " and p" , Since 01 and Oz differ only in the interchange of Pa 3 they give the same result after the integration over momentum space, so that only one need be considered, and we obtain for ~ ~ since p 1 • k1 = E E1 ~ -Pl • -k etc. °' is a polar angle measur- ing the location of the vector -p 3 with respect to a plane taken for convenience through Pa and k. (v. Fig. 4.2). I\ All the re~ ,.. sults of section I can be used by replacing Pl by P2 and p 2 by -43- The angle between !).-, and P-:;;u is ~J and k is ( 4 • .35) e ' while that between p-:;: kJ r · Then we have, by spherical trigonometry, P3 • k = P3k (cos e co~ r + sin (} sin 'Y cos 0(. ) \ rl \ \ \ \ I I -k (Fig. 4.2) 2n = 1:k f [E.(T-E2.~-p r1. E E3 + k 0 Zrr [ c. (T-El. )-rz.k c::os y {(E. E1 + r3 kcos Q cosy}~- (r 3 ksi"'0 s;.-y),_ 2n ( T - E2 - r2. os y y(E3 .:os y + f>3 c«>S c:. e)2. + M' sin'y where we have used J1 ::. 0 to replace £. by k -::. expression is now to be integrated on -I]_ _ rJ y£ ,_ _ f . ._ . This EA ' E3 ~ E~, where we = have, by (1.06), (1.16-.15), with m1 ~ m2 m3 : M, 2 (4.37) P2P3 cos 9 i(T 2 + M ) - TE2 - (T - E2 ) E3 = and E3 ~ PzR (4.38) E~ + (4.39) P3 ~ p~: (T - Ez)R, p~ - P3 ~ Pz E3 : T - E2, E£ - where by (1.07), (1.16), (4.40) R-: 2 :. T 2,..0M 2T -44- Let (4.41) X = E0 cos y .,. f .3 cos 0 Then it can be shown that -p 2 dx (4.42) T - E 2 - . P2 cos 'Y with the limits of integration x 11 = E3 (4.43) x' (4.44) yx 11 1 cos y ~iV\ 't. "2.. x ~ x', where 11 ~ E 11 E 3 I+ + M"l.. c:. :"".,_ y .rx"2.+ y i M 1Vx p 0 •, x 'T ~ 3 r'! cos y r y - E 3 '' - p 3'' cosy x' ( 4. 45) x'' cos y - p 0 11 I dx '2.. + M~ SiVI "Ly I oj E 3 + p3 f I+ R -1 I oj -== 2 tv.vil.. R '' p II E -3 - , 3 t -R where (4.46) by (4.09) with ~ : O. Now put these results into (4.34), using (k 0 - p2 ) 2-r-1. = -2(E 0 E2 - M2 -" ~- r1 - Pzk cos r ) dJl ::::: 2 S y d y . (4.47) and ( 4. 48) ... ~ _ TT iY") j6 JE~ - 8 L c kT "" 1" l d p4 2 lG..,..k_, R- R) E2 E - o - C 2. - M 'Z. -f s~n y dy zI r - p l~ c oS y l. 2.. 0 Before integrating over 'Y , we do a partial integration on E2 , to obtain (4.49) Integrating over y , we giet -45- IE; == z ~ck.T df':,_ { b ( 4 . 50) c! M 2 E.- E,_) ( M lE.-E~) +~r't,J {to."'k- R - R) [M-z.(E 0 -Ei.f'+r-1.(EoE'--M?..)+~r~-~ 1 Inspection shows that at E2 = M the integrand become infinite with l / p2, s o t hat for very high energies we can approximate (4 . 50) by putting E0 (4.51) r:5 - = M in the integrand: jb M (Eo - M) --'-~-- Z~ c. k. T and cos h- 1 x ,._, log 2x for large x, we can estimate (4.52) c! ~ by ~ t..c·[~._M.,_d I + i~(:,-M)J (loj ~ - I) loJ ~ ~ using the transformation formula to obtain the energy E rr of the incident meson in the laboratory system: (4.53) The graph, Fig. 4.3 at the end of the text, gives va lues of ~ obt a ined by numerical integration. It is of interest to ascertain the effect of the inclusion -46- of virtual neutral mesons on the result of such a calculation. If we do this, we have tvw more diagrams to consider. They are given in Fig. 4.4 below. p (fi) p lf3) ,.. , ____ s I\ r, p-( f1) ", ,.. , s ------ rz __, -Pl Fl> p- lr.> P( r1) I \ I \ I \ \ " N ( ko) \ " \TI+(k,) <~> I I ,.. (b) ,... / n + l k,) N U<J (Fig. 4.4) Each diagram represents two processes, the second being obtained from the one shown by i~erchanging ~2 and a3 everywhere, replacing primes by double primes. ,.. ,.., (4.54) rl :::. ko ... "k1, (4.55) " SI " :. " - k1, " r' P2 2 " "' + P31 = Pl " r" 2 = I\ " - k1 P3 " + "Pa s" :. Pl J\ Remembering our rule about signs in the symmetric theory, we get for the matrix element of H in this case (4.56) 3 2 HN = (4 rr ) / gg 2 ( Ze 0 )-t · z-t(o3 - o4 ) where (4.57) 03 ~ f(p3Jys-tlp,) ( s' -r'l.J- 1 . 2 ( ~ lrl.) y'i (r, - M )- 1 Y~ t l~o }- f lr2) y) l r,'-M r'11s-t (k t lr1) 1s- tlr,} f lr2-l 'k, t lko) · ( s, r-'\. r , [ ( ~ M l -• + < ~i' l_ 2 _ 4 l. _ 0 ) ] ::- J) r,1 '1. l -, and where o4 is obtained from 03 by interchanging p2 and p3 in all terms. Then the cross-section is proportional to -47- ( 4. 5 8) I ~I \ 2. I Hc + H N I ::= 2 £ - I ( 2 Tl ) l ( ~ C ) ~ 'j 2 Ij 2 1.. ( 0 I - Oz. ) + jo'2. { 0 3 - 0 'l )f 2 tivistic limit and substituting into (4.10), total cross-section, neglecting ( 4. 5 9) r2 << M2 , rr ( 3j ~ + 2 jco 2.) '2. ,) ' 6 :::::- -----..,..-----=-2" 3 't/,_ l t c } l = 2 , the cross-section -2 so that in the Kemmer theory, with ~ 1 ~g is 16/9 the value obtained by including only charged virtual mesons, in the non-relativistic limit. We should consider also the effect of negative mesons colliding with protons, so that antiprotons might be produced in the reaction (4.60) Again considering only charged mesons in intermediate st a tes, we now have t wo diagrams, which correspond to the interchange of the two protons entering the reaction-- the initial proton and the proton going backward in time which represents the antiproton produced. Plr1> P r" , lp1) ,," "-II r ;' -~----- / - - - -- I I lb) I I " /n-(k,) (Fig. 4.5) P( "ko) -48- " " " " + k1 = ko "" k1, "p" = -p2 " " " + ko ql = P2 PN' q2 :. -PN p/\ I (4.61) /\ ( 4. 62) /\ /\ 1- ,.... (4.6.3) t (pl), ,..,P2 t (pz) ,.. t (pN)' ko t <ko) t (p1) :. M Pl ,., PN t <PN) '::. M :. -M :. M f' (p2) t (ko) Then the matrix element can be written as ( 4. 64) H = ( 4rr) 3 h J3 i ~ c 1 ( 2 £ )-f ( O, - 0 i. ) where (4.65) o, - f{r"') 'Ys-~ t,,J (1,2-r).J-'. \¥ (p ys- ( p M ) - 1ys f l ko} ~ (f"') is 4' l r ~ F• l k, t ( k (1,'-r'l.) l fn--M~) t (r~) v~ t ( k J ( ,,2 - r i-;-' · f tr,) rs- !r''-M )-' r~ tlr,J::f (pH 1 'Y!; tllo) f tr,) k, ~(fi_) 1 1} - l. ) (4.66) 0 ( ) 0 ,l. -r " ( ._ '") ( ,, h 111. _ M '1.) (4.66) is seen to be the same type of term as (4.15), while , (4.65) is new. Here we must omit the factor 2-~ which appeared in (4.1.3) because the final integration over momentum space in this latter case includes only one of the 11 identical 11 protons, the antiproton observed, while in the former case, both the protons are observed as produced in the reaction. Then the cross-section is given by (4.16), with an additional factor of -49- As in (4.20), (4.69) l. 102. \ ~"L 7- [ 4 c Pi.. k,) ( f,. kI) - ,_,. L ( f -t p,) J 4 Met ( 1i.i.-r--i.)~ ( f"l.~ M-i.)-i. 2 I 4 A long spur calculation shows Proceeding as before, in the non-relativistic limit, we find for the total cross-section, neglecting }1 2 <~ M2 cf' ..:.. (4.71) - (~f (~f (*f ?rr 2 S" • 3 't/ z. This is 14/9 the cross-section (4.29) for the process (4.01). For higher energies it is again not possible to carry out the integration over momentum space except by setting p ~ 0 and neglecting the first term of (4.70). This latter approxima- tion introduces a more seri ous error here, however, for even in the non-relativistic limit, the term neglected contributed 2/7 of the total cross-section (4.71). Hence our final result can be expected to be in error by about this amount. (4.33), the crosa~section As i n reduces to (4.72) The second term of this will give the cross-section (4.04) or (4.50). The first term is smaller by a factor of 1/10 or less. f If we c a ll the contribution of this term (4 . 7.3) 6 I - j{. 8 "kc k.Tl. E,o M Pt E,JE, E - E: 0 I obtain -50- Numerical integr a tion then g ives the follo wing va lues, tabulated aga inst the energy Err of the incident me son in the laboratory system: E rr /M 6', ·(~J-3(~ )-~.1oq 6.1 12.0 34.5 52.4 100.5 2.75 7.80 7.09 6.22 5.69 We can thus conclude that the total cross-section for the production of antiprotons in the collision of negative mesons and protons according to (4.60) is of t he same order of magnitude as the cross-section for the positive-meson-neutron process (4.01), and in later wo rk we shall take them equal, using the more precisely known cross-section (4.50) for both. Inspection of the integrand of (4.34) shows that it is largest when Pn,;:, is in the direction of k. Hence the protons are emitted mostly in that direction, and the antiprotons mos~ ly in the direction of the incident meson. -51- V. PRODUCTION OF ANTIPROTONS IN NUCLEON-NUCLEON COLLISIONS -- PERTURBATION METHOD (NON-P~LATIV I STIC) We shall now calculate the cross-section for the process by which a proton and a neutron collide to produce a pair: (5.01) N(a1) + P(q2) ----"> P-(pl) + P(pz) + N(p3) + P(p4) We assume for simplicity that there are on l y charged mesons in intermediate states, so that there are two possible diagrams (Fig. 5.1), the one being obtained from the other by the inte~ change of the two protons entering the reaction- - the orig:in al proton and the proton moving backward in time which corresponds to the antiproton produced. Pl~) ,,.. I r, ----- p-(f.) Plf.. ) s"' , ,.. ,,.. ri t N lfi) - - ---- ---- , ,. r1 ~----.-~----- ~b) l <>. ) (Fig. 5 .1) In the center- of- mass system, we have (5.02) with the Dirac equations cli t (ql) :: M t (ql)' q2 t (q2) = M t (q2)' i\ 'f (P1) ~ -M t (pl), pi t (pi) = M t (pi)' i-::. 2,3,4. T is the total energy in the c.m . system, and T = 4M is the (5.03) t hres hold for the reaction in that system. - Let E be the energy -52- of one of the incident nucleons in the l~boratory system, L e., the system in which the other initial nucleon is at rest. Then the Lorentz transformation gives (5.04) and we find that t = ?M is the threshold energy in the labora- t ory system. Because of the number of variables involved and their complicated relationships, we have not been able to evaluate the total cross-section for this process by integration over the momentum space of the four particles produce d, except in the limit in which these particles have kinetic energies small compared to M. Then the energy E of the incident nucleon is only slightly greater than ?M. The absolute s quare of the collision matrix element H reduces to a constant, and the total cross- seotion is g iven by (5.05) where given by ( 1. .33), and can be written (5.06) /OS- . 2,..{a 1 using T - 4M::. U :!: 4-"(E - 7M) for U << 4M. In (5.05), v is the relative velocity of the initi a l nucleons in the c.m. systan, and it is given by (5.07) v/c = 2q/E 0 The factor 4l in (\5.05 ) comes from the average over the spins of the initial nucleons. -53- The matrix element can be written down from the diagrams of Fig. 5.1 in the usua.l mc<..nner, and we obtain (4rrj'1l'cl.J2. ( }x_ (0 1'-o,'') -~ (O~ - 0 i."J] (5.08) H = -- fl ' (4 lf J ~.iL. ,_)'1. {o'-o"-02.'+02..") ·n,, C I I J (5.09) From diagram (b) of the same figure, (5.10) 02 '-)-1 . = t r" Ys-'f'.1.( i' )("''2. r, -r I -( ) 'Pl~ .. ) 'Ys ( t'-MJ-·r~t l;~J (~2.-r~J-'f{r3)y~t(r,) or, o~ are obtained from 01, o~ by interchanging S2 and B4 in all terms. From the diagrruns we also obtain the relations be- tween the four-momenta of the particles involved: (5.11) ,., "' ... " "' =. P3 - qz, r3 = P1 r2 - Pz' P3 rl ,.. ,., " ... rl -- P4 .,. r2 = -P1 - P2 .,. ql : -P1 S A I :::: " ql ,., t I A A A A. /\. A, A. t" I -::, (5.1.3) ys-+<,.J ~Hr.,J ~'tf1'1.J f(r3) ys- t(r.> 0'2..' = ~ {ri.' r 2. ~) ( t" ' .,_ M"' ) ( "' ._ "l..\ I\ , \.. r, r - - after some simplification. r-3 - r) In the non-relativistic limit, the *(r y * f Dirac equations for the final particles become simply ( 5 • 14 ) y'I t (f 1) =- I ) \ 'f l i ) -= + t l pL ) I l ~ 2I 3 /t ) -54- while for the initi a l nucleons ( 5 • 15 ) E ( 5. 16) ( 2 IV\ ::. 0 q .:, M-{3 2IA , 'Y 'f - , 'Y' ) t {i I) = M t l , ' ) { 2My.. +jy,J tliz.): fv'l~{'ll..) if we t ake the x-axis in the direction of the incident nucleons. In addition we have - r, (5.1?) ( 5. 18 ) ","L "\.. r, - r I = 'y' Y~ - 1 Y• 11..1\ -;: ; r2. - r ,.1 'l. = - M1. ( 2I + r '-) r = 4 M -r s M ._ = - 4 M f ,.. - M ~ 8 M ,_ '1. ,..i r3 '"2.. "l... - •"L - -= 'l.- ) flr"i) r.'+lr,) = ftr-tJtMy.,-1y,)t(r,) = -1 f l p•d 'Yl ~ l r I ) since by (5.14) 'fl1~Jy-1ilp1):: flf'i)'4-'[r,)=-tlr.,J+-lr1)-:0 (5.19) <5. 20) '*A, J::; \\i l r r:' * (1~ J -= 'f-1 Lr.,) { M y~ -1 y. J f (r..) ( 3 Hy.. - M ) +r , 1.) :::: 2 M f l f" ) t l i 4( ) 2. 'L ) since by (5.16), _i'Ya tlii.)-== (2.M'}At-M) tli'l.J and by (5.15) + lf'i) "{'f -= f lr'f) (5.21) .'. o' =qt (pz_)ys- tl1,) ffr'ily.t(r,J flr1)ys-+c,l.; ' (5.22) ~ 4M'l. t2M'l.+ r~J7.. ~lr.,Jt l ~"l.)i(r~1ys-t<r·J 0 2. ' = _ ~lr,Jys-tl,,J 4M ( 4M'l..-r"1-> (2M'\.+r"1.J L 1flrl.l Ys- +l 1,>12- == 4 'Ml.s-r'f1.+M)ys-lf.+M)ys-= - i'ML l 1, - rL ) = + ' -= L \ f cr3 J rs- '} l 11. JI~ (5.23) 'l. ,... 55- (5.25) . z =-4 '.;r \01' 12. - L l fl r'i )t (11.) 11. ={4 M ( 5. 26) "l. ) - ' sr ( r., + M ) ( ~~ -t M) -= (2M2.)-' ( f'i+ i,) 1 =- 3 (5.27) ,-( L '\' r 0 <5 • his) ..• 3 ) 'Y s- t (r, Jl2. ~ - (p,+p3) 4 i. M '2. -2.. - 3 l i o '\'1. ~ L~I 0 i. "l'L-_a_M_1.._c_2M-'2.+_r-_'\._r--c-4_M_'2.._~-r-~J-'l.. 2. Similarly the cross-product terms can be evaluated: ( 5. 2 9) cs.3o) L o, 0 I,, = ~ L I 0 .' I 2. I L. o,'02.' = Io,"oL'' :::--3 [H,M'l.(2M'2.+r~J3(4H'l.-r"\.JJ-' - (5.01) - _!_ ~ I oi_' IL. 2.. L., (5.32) Combining all these and multiplying by M4 /E 0 (-E1 ) E2E3: -t, (5.33) <l'rs = J's- VJ F ( r( Mr~ u: r( Mr2 ~, ( ~:r ~ = 'l. 't 0 • io-J ~ E-;,..1 E~ w = c "" ... where The case of scal a r coupling is obtained by replacing )'sby 1 in (5.09) and (5.10). Then in (5.10) the factor f (p3) t (PJ! -56- wi ll give zero because of the orthogonality of states of positive and of negative energy in the limit of zero momentum. Thus only diagram (a) above contributes to the cross-section non-relativistically. Working out the spurs in the same man- ner as above, we find 4f ( 5. 34) <S'sc. = 3 3 S" {3 L "2.. 2S" ( _r_ ) ( M"'c: ) 1'c 2- T - ( c. -1 M 1/ .,_ -~ ~' t-1) 2( l-+ 2M1} which is about 12 times the pseudoscalar result if the same coupling constant g is used in each. In the case of pseudoscalar mesons wi th gradient coupling, {5.09) becomes ( 5 • .35) "t..)-f f '-I QI I :=- ~- lrl.) 'VS"" ,_., rj t (j I )("'' rl f (r'"i 1y s- r:i. ( s '- M J_, ys- F. ' +- l f • J · r ( . r"-)- '-f (f 'VS r2. +(, J. ~2 2 - 1 ) I 4 Using (5.11), we can rewrite the middle term as follows: ,. _ l s -tM l Ysrl,__ I 'f l ;° • } ( s -M rs ..36) t- (f'i ) r~r?.. = - (\r'I ) r1. l s - M r, ~ t f ("'<.. "\..,-} + .s - M == +t1.iJ l s- ' - f s- ' - M J l s- ' + r· )t l r. J Ls· H ... J_, = ~ t f 'f ) t s M ) 3 ~ tf ls M ~ t r'I ) [ t s- • + 3 ""' 2. J s- ' - rvt ( 3 s• + M .. ) J ~ cr,J( s· M~J-.' 2 Non-relativistically, by (5. 18), a12 + .3M = 0 and 'f' (p4) f (P1) 1 ""/'L I ) ,._I "\.) - I ) I 'f ) ( I - "l =0 "l. - I ) IL - "1. 'l. } _ , ':::" L_ (orthogonality of positive- and negative- energy states), so that O{ = O, and only diag ram (b) of Fig. 5.1 contributes in th i s case. This gives -57- *l (5.07) r'I. oi.' == ftr,)'Ys-r;'~t,l) l;,'l.-r"\.)- 1· f ~ ) ys r-3 ( i, - M }- I 'Y$ r, I t ( 1"2. ) • * (G._ - ~ ._J-' Lr J ys- F~ t (r, ) = -c2M)~ f lr"l.) ys +t 1,} ftr~) tt'-M) 'f('l2) · Fr1)y.+lr· 1 t ;,·"'--r''r' 1V'-H"'-r' t fi -r'r' 3 1 2 Where We have used q: l r ~ ) vs t 1, - r t l rL.) y S r. I~ 1 L. ) ( *i ~ , 1 ::: I ) ::; q> l / -i.) y s- l M - Fi- l ~ [i • J ~ ~(ri.JlM+pl.)y~tl 1 1 )-== 2M *(r,_)ys-'.f(~,J, etc. Working out the crass-section as before, we obtain ( 5. 08) 6 _ _I_ . 2 - ~' ( ~ ) .., ( 2 rsv - 3s-{3 M (5--=) 'kc. '2.. ( E- 1 M M M) 8 . ~ )l ( + ? )- 1_ ~ )\ 2.M'l '2 ( 4M'l. 2 · -58VI. APPLICATION OF THE WEIZSACKER- WILLIAMS METHOD. The Weizsl!tcker-Williams (W.-W.) method was first applied to problems involving the electromagnetic interaction of fast moving electrons 15116 • If an electron is moving with nearly the s peed of light, its electromagnetic field is compressed in the direction of motion because of the Lorentz contraction. To an observer this field appears as a pulse of very short duration, and such a pulse can be represented by a certain distribution of quanta moving in the same direction as the electron. Thus the interaction of the moving electron with a given system, say an atom, can be calculated from the previously known interaction cross-section for high-energy photons with the system. Many examples are given in the paper of Williams 16 • The w.-w. method was first applied to meson-nucleon interactions by Beitler and Peng 17 in calculating meson production. We shall follow their treatment in developing the necessary equations for the case of pseudoscalar mesons, in order that we might understand the approximations involved. mesons, the source-free field equations are (6.02) 0 -t' /'at = fA t f r - -CV'}( (6.03) () ~ /()t. (6.01) == (~ For pseudoscalar =c 1) r i' where f is the mass of the meson in these units. combined to give the Klein-Gordon equation (6.04) ~ These can be -59- + and for each component with a corresponding equation for of r . These have plane-wave soluti ons of the form where (6.06) The density p and current S of mesons are given by p ;::. ( Lf n r' ( r *. r + <P * <P + it ... '1:) (6.07 ) s = ( 4rr) _, { ~ • r + ~ r *) satisfying the continuity equation \J . s +~ ()t (6.08) 0 The meson field of a resting nucleon, which is treated as a point source of mesons, is given by ( 6 • 09 ) f '1( I ::: .f ( 6 ' VI ) cp l r 1 ) > =gp /2M, where g is the usual coupling constant. spin vector of the source nucleon. ~ is the (Q,uanti ttes referred to the system in which the nucleon is at rest will be denoted by primes.) By (6.01) i:>' = O. Now consider a system in which the nucleon is moving in a straight line with a speed v. Since r and cp together form a four-vector, we have ( 6 .10) or, since (6.11) \'.[-here rJ( -:: y ( r lt p' = 0, / + V 4' 1 ) ) cf :: }' ( <i I + \I rX I) -60- We seek the field at time t at a point P a distance b from the nucleon path. If the nucleon passes (v. Fig. 6.1) vt b r' p (Fig. 6.1) P at time t : 0, then by the Lorentz contraction rule, r I = ( 6 .12) since x' = y x y +b X / 1. L.. =. j 'Y V 1: + bi_ 2. L I.. = y vt. By ( 6. 00) , ( 6. 09) ' ( 6. 11) ' 'j/ I o""l.r' _ x o'dx' = - rv -µ- 1 rx (6.1.3) i d 'f{r'J -<! )( +6'}?b] [ at 'dx rv a f t rv -~ ~, -a 'dt 6' x [ f-l v d (} t -l- 6'} where r' is expressed by (6.12). ;b] cp(r') The meson current in the x- direction passing the point P is given by <s . 14) sx = 4~rr ( cp • r )( + <P r x• ) = .J__ v f rt I ' 2rr Now we make a Fourier analysis of the meson field observed at p. (6 .15 ) Let J -+ao <p(r') (2n)-II l e t(J t ~ ( w ) aw -co where -61- <6 .1~) ht..., l -_ P-· ri f ,,.,1, '"e. -oo = -'K ( r b s e c o<. ) yv VFI rr o > w .. +h t CJ."' 0( = w · YI"' v To prove this last formula, make the substitution y V -l: =b Jt ~v cosl..l r-(0() J 'f ( 6 • 1 7) S jl'\ h ( <p- t ol ) J r I = b C .S k { 'f - l oi ) 0 I Then we get h (w ) - -+Go .. l'cl \ln)-i ~v J e•p (-i ~: s;,.l.. ('f-c"J7b <•sl.l<f-<"•l).Jf (6.18) \j Y ~ 'i - DD +icl +oo -t&llt e,r[-rb seco< cosh'f] d'f -oo+.:oc rr if we pick tan r:J. ::. W/f'Y" so that the terms in sin h exponent vanish. long the real in the Deforming the path of integration to 1 ie a- ~ -axis, l<' o ( } ( 6 • 19 ) f we get the standard integral form 1 ) == 2 f-+ oo _ 00 e _ ~ co5h ~ 'f d f This gives (6.16). l rryv r yv J Substituting from (6.16 ) into (6.13), r (6.20 ) I)( -:::: r I 6' ~ -i. f r- +ao -ao W '- e it.> f; J..( o (r b Se l. r:/.) J W +oo _ «s- i w sec o1. e '"' t 1-<' .' ( r b sec"' ) J w -(p The integrated current in the x-direction past the point P is t -hen given by -62- where we have used the theorem that if g( w ) is the Fourier transform of f(t), f (f(t)J Jt +oo 2 (6.22) -oO If we now integrate over all impact parameters from b = b min . to b : oo , where bmin is some minimum impact parameter to be determined later, we find that the total momentum in the pulse is (6.23) and -63- Now assume that this pulse can be represented by a distribution of mesons such that there are q{ "" )d c.v mesons of energy If we assume the energy En =y M of the nucleon is very large, then we can take w > > ~ , so between w and w +cl w that each meson carries a momentum <.J in the x-direction. Then the total momentum transfer is approximately (6.28) (6.29) where we have put v : c ~ 1. Using f -::. gp./2M and En = 'Y M, we can write this as (6 . 30) 1( w) d w = 21n ( t~ ) wE: ~ l 2 l k' K• - r .. b,_ ( K , '" - t-< 0 0 2 JJ putting ( 6 • 31 ) b = lo""':\'\ ) Now i f a high- energy nucleon interacts with a given system4 the cross- section can be found by first calculating the -64- cross-secti on for interaction of a meson of energy w with the system. This is then to be multiplied by q ( w )dw and inte- grated over w , to get the nucleon interaction cross-section. In this it must be assumed that the nucleon is very little defleeted during the collision, since we had t o take the path of the nucle on as a s t raight line in the above derivation. Hence the method is applicable only for nucleon energies En >> M, the nucleon mass. The spectrum q ( C.J ) d w holds only for we had to assume the meson momentum fW"'l..-r'l,.:: w (6.28) . t..J >>r since in setting up On the other hand, we do not expect to have mesons in the spectrum of energy comparable to or greater than the nucleon energy, and indeed, the interaction of such mesons would cause such a large momentum transfer to the nucleon that the path could no longer be considered straight. off the spectrum at a meson energy of a fraction less than 1. taki ng w - - Hence we must cut Q E j.J " where p is We shall follow Reitler and Peng in fl - t. It is also necessary to pick a reasonable value of the minimum impact parameter b . rest, take b : ~c/M , For a collision with a nucleon at the Compton wave - length of the nucleon. The momentum transfer must be much less than M in order that the path of the moving nucleon be considered straight relaUve to the position of the other nucleon, so that we then see from the Uncertainty Principle that b ,.., 6 x ~ 1ic/M. Since then we have bp << 1, the quantity in brackets in (6 • .:30) can be shown to be roughly 2 log 2/Cz, where C ~ Euler's constant= 0.5 772. T nus q ( w ) depends on the impact parameter b only logari thmi c ally. -65- Of course, with all these a pproximations and uncertainties in the various quantities b, (3 , etc. which occur in the ex- pression (6.30) for the meson distribution q( CJ ), it is clear that we can expect the method to give no more than an order-ofmagnitude estimate of the cross-sections calculated therewith. Our results may be in error by as much as a factor of five or ten. It should be noted also that we are using the theory of a neutral meson field and neglecting any effects which may be due to the change of isotopic spin of the nucleon. We are forced to assume that the distribution of positive mesons in the field of a moving proton, or of negative mesons in the field of a neutron, is given also by (6.30). We shall now apply this method to the production of nucleon pairs in the collision of a fast moving nucleon of energy E with another nucleon at rest. It is assumed that the mesons in the field of the moving nucleon beget nucleon pairs on the resting nucleon through processes like (4.01) and (4.60), and the cross-sections for such processes a.re taken as that calculated in the first part of Section IV and expressed in (4.50). Call this cross-section, as a function of the incident meson energy Err '=- W , 1 6 (w) • Then the total cross-section for the produc- tion of a nucleon pair in the collision is fJE ( 6 • .32) 2 J4M ~'(w),fw)clw The upper limit of integration will be taken as PE ~-E in ac,.J cordance with the above arguments, while the lower limit is (..) : 4M, the threshold of the meson-nucleon process. The fac- -66tor of two occurs because we must consider the two Lorentz systerns in which first the one nucleon and then the other is taken Write ~' ( w) as initially at rest. as 6'', w ) == (-fr (~ (6.33) r f(,,,) where f ( w ) is the quantity plotted in the graph of :F'ig. 4 • .3. Put (6.30) in the form 1l ( 6. 34) J t.J ) (A) = ~Tr \i ~) ~ ~ ~ ~ (~ ) where G ( ~) = 2 ~ Ko [1 } t-<, l l) _r' ~ 1. K( ( 1 ) ] 1- - [ Ko l ~ )) i_] · (6.35) f[ }=~(~)'+l~t (6.36) x:. w /E. with a::. u/M =. 0.156, I ( 6. 3 7 ) .. . ={"...'-+.'- d E ) "" ~ (fc r t~ rl 'Ii. x t ( b ) G( } , " " ~M E G(z) is evaluated from tables of modified Bessel functions, and it is tabulated below. x 0 0.05 0.10 0.15 0.20 0.25 0 • .30 G(z) 3.017 2.993 2.930 2.783 2.582 2 • .362 2.142 x 0.35 0.40 o. 4-5 0.50 G ( z) 1.933 1. 741 1.566 1.408 The 1ntegration in (6.37) is carried out numerically and the results are tabulated below. E/M 6 13.3 (~)-'t("~~r··l0 1 0.14 16 20 26.7 40 80 160 0.29 0.53 0.87 1. 32 1. 76 1. 73 -67- VII. PRODUCTION OF MESONS IN NUCLEON- NUCLEON COLLISIONS For sake of comparison it is of value to estimate the cross-section for the product ion of mesons in nucleon-nucleon collisions. (7e0l) We shall consider the process N (k 0 ) P(k1) ~ N(p ) + N(p 2 ) + n-t(q ) 1 + This calculation has been carried out by O. Merette 18 , but her method of momentum-space integration is so complicated and di~ f icult to understand that we shall repeat it using the methods and formulas developed above. The pertinent diagram is given in Fig. 7.1 below. N l ft) N ( f•) s" I I ---~-~------- P(k,) (Fig. 7.1) From this we can write down the transition matrix element (7.02) H ( ~ j!; ~ ) 3 ( 2£ )-1. z.-1: ( o' - o") where o' (7.03) t lr I ) 'Ys- ( r ' - M J 'YS" t ( ko) . . ( s''-r't. )-' f (r2.J ys- t{ k, J and 0 11 is got by interchanging (7.04) -I p1 and p2 everywhere. -68- In the center-of-mass system let = (E 0 , ~k), Pl: (E1, p1 ), .JJq :: (E 2 , 'P ), iq = ( s , q), T = 2E 2 (E 0 ,k), "k 1 (7.05) ~ 0 ~ We can simplify (7.03 ) to obtain o/ lr2-) y~ tlk,) 'f>{.r,)1 t (l<o) ( s'1.-r J ( r' lV\~J o' (7.06) 4 4 - The differential cross-section for the process is given by (7.07 ) where we have a factor tial nucleons. (7.08) t to average over the spins of the ini- The relative velocity v is given by v/c: 2k/E 0 while the density of final states is, by (1.20 ) , f F -= l 2 IT~ c. )- & EI E2. £ d E 2.. cl f2 2. d £. d cc. (?. 09) where ~ is a polar angle locating the vector ~ and measured from the plane through ~ 2 and i. Combining (7.02) and (7.07- . 09), we get (7.10) j~dEi.JJlz.c:le.do( Mq[fo'-o"l2.. lbrr'-i;(. k E 0 where we have already multiplied by the factor M 4 /E~E 1 E 2 which corrects the spurs for normalization. Evaluating the sum of the absolute squares of H by the same methods as previously, we again obtain from L0 0 1 11 an exchange term proportional to A ) 2 pA - Pa which we are obliged to neglect in order to carry 1 out the integration. Since we are interested only in high~en- ( ergy collisions, we may also neglect the meson mass p· we find for the total cross-section Then -69- J( 3' ko · ~ ) dE2. dJl 2 d£ Joc (1.11) 6 == tb..-'t-okT ti'''-- M'") ( s•'--r') j~ {E E-?j·k}dE,__Jf2z.d£dOL 3 2. rr i.~ c k T 2. ( T - 2 El. ) ( E 2. E 0 + pz. · k - M ">- + ~ lo) J - r 0 The first integration is over all positions of ij, which lies on a surface given by (cf. (1.06)) p.-,. Cl = P2q cos 9 -- 2iF2 (7.12) k; 2 , - y ~T(T - 2E 2 ) = G2 - M2 F2: T2 - 2TE 2 (Fig. 7e2) Let the angle between P2 and K be (7.1.3) - (T - Ez) £ ' k r. Then Fig. 7. 2 gives q·'iC = qk (cos (} cos y + sin IJ sin y cos ol Thus the integration over ~ is easily carried out, after which we integrate over E. , and use the relations (7.14) (cf. ( 1.12- .1.3) ) . 6' _ (7.15) - Integrating then over y , we obtain 11.T dE'{T-2E'J . lb-fie.kl. (T'--2TE'+M'),,_ J" M since the maximum value of nucleon energy is given by (1.16) as -70(7.16) for T - 2M >:> p. We note that if . En is the energy of one of the nucleons in the Lorentz system in which the other is at rest, T2 ( 7.17) = 2M(En + M), k 2 :: !M(En - M) (7.15) is inte g rated numerically, and it g ives the following typical values: En/M 5.5 12.5 19.5 41.3 80.9 161 6·( .i~ )- (~ r~IOz. 1.37 0.97 0.80 0.47 0.30 0.18 3 (7.15) is quite well represented at high energies by l, ( 7. 18) cJ' ..:_ ( b ( c M ["' ( I 0 j 2 1~"' - I ) It appears that meson-production and pair-production become of comparable probability at very high energies. -71VIII. PRODUCTION OF ANTIPROTONS I N COSMIC RAYS. Using our calcula ted cross- sections we can now estimate the numbers of antiprotons produ ced in the atmosphere by cosmic radiation. Consider first the interaction of pi-mesons with nucleons to pr oduce pairs by processes like (4.01 ) . Since the energy required is so high, we can neglect the effeet of the nuclear binding, and we can use the cross-section expressed by (4.50 ) . This will be written as (8.01) where f(E ) is plotted in Fig. 4.0, and E is the energy of the incident pi-meson in units of M, t he nucleon mass. According to the work of Sands 19 , mu-mesons are produc ed in the upper atmos phere with a spectrum of the form E-y dE, with y - 2.5, for high energies, and we can take this f orm also for the spectrum of pi-mesons in the atmosphere. Thus t he proba bi li t y tha t a pi-meson wi ll have energy between E and E + dE is given by (8.02) where Ec is a cut-off energy on the order of ~M . We sh all consider loss of pi-mesons only by decay, neglecting nuclear interactions. The fraction of mesons which survive a dist a nce x in the atmosphere is g iven by exp(-x/c L' ) : exp( - xr /c "'t: E), where 1 'T :. (E/}1 ) L is the lifetime of the me- sons corrected for relativistic ti me-dilation . -r is the lifeti me of the meson at res t, approxi ma te ly 2 . 6 x io- 8 s e c. 20 If -72- p(x) is the density of nucleons at distance x, the probability that a meson of energy E vvill interact before reaching sealevel or before decaying is = (8.03) I. Hr( e - Xr-/c x) 6( E. ) J J( • i: E. H is the distance of production of pi-mesons above sea-level, fo exp ( - ac. (H - x)j , where r = 7.8 x 10 2 0 cm- 3 is the density of nucleons at se~ level. ()!..:: 1.4 x 10- 6 cm- 1 • and r (x) = 0 If we assume mesons produced at about 100 gm/cm 2 below the top °' H = 0. 1; and H -::.. 16. 4 KM. ( 8. 0.3) becomes po 6' ( E ) e - ot H ( l - e - ~ H ) ~ - 1) of the atmosphere, e(8.04) C'-~-Ald - er E c""' ~ -I Thus the probability that a pi-meson will make an antiproton in the atmosphere is given by .1 3 H( ,~ ) (8.05) y: ~ g(E)p(E)dE =2 roe t.c. (l\M )t. . iM ~y-1) E/-' f4M E -y fl E) l l -e -iHJ ~-· d E, 00 1 -ot C oo where the factor t occurs because a positive meson produces an antiproton only in interacting with a neutron, a negative meson only with a proton. Numerical integration with 2.5 gives a value of 92.1 for the integral, so that (B.06) Y = 2.4 x 10- 6 (Ec/M)l.5 (g2/fic) 3 or about lo-6 ( g 2 /!1c) .3. iv * We leave all results expressed in terms of the coupling constant, since its value will probably be revised when current calculations of the nuclear interaction p otential taking into account exchan%es of t wo mesons are completed (R.P. Feynman, class lectures). It should be remembered that we use unratio~ alized units, in which the Yukawa potential is of the form g2exp(-pr)/r for the scalar theory. In pseudoscalar theory, g2/flc is of the order of unity. -73- Vie can also calculate the production of antiprotons by the primary cosmic rays. Since the energies are so high, col- lisions can be considered as between individual nucleons only, neglecting nuclear binding, and we can use the cross-sections esti~ated in Section VI. The spectrum of primaries can be taken as that in (8.02), with a cut-off energy determined by the latitude. For a lati,..., tude of 41°N., E 0 4. 7 Bev. or 4.4M. In penetrating a dista.nee equiva.lent to d gm/cm 2 of atmosphere, the number of anti- = nucleons produced per incident primary nucleon is Y-:. Nad (8.07) I 00 <! (E)p(E)dE 1M where Na= 6.02 x 10 23 is the Avogadro number, and d (E) is the cross-section calculated in Section VI, as a function of incident nucleon energy. tain, with (8.08) Numerical integration is used to ob- y =- 2.5, 1 5 4 Y =- 5.3 x 10-9d(Ec/M) • (g2/1lc) = 4.9 x lo-8ct(g2/110)4 Thus, considering that a primary nucleon penetrates to an average depth of 100 gm/cm 2 before losing most of its energy by meson production and other processes, we see that we can expect only about 5 x l0- 6 (g 2 /fi.c) 4 antiprotons per primary cosmic- ray nucleon. It is instructive to compare the effectiveness of different parts of the primary cosmic-ray s pectrum for antinucleon and for meson production, by evaluating (8.07) for different cut- off energies, using the cross-sections calculated in Sec- -74- tions VI a.nd VII. Carrying out the numerical integrations, we obtain for the yields Yn of anti-nucleons and Yn of mesons, per primary nucleon, the following values. (g 2 /'bc: 1 and d = 100 gm/cm 2 here.) Yn Yrr 1000 4.9 x io- 6 4 2.9 x 10- 5 4.7 x 10- 1.9 x lo- 5 6 7~0 x io- 1/60 1/1.1 2.8/1 4 • .3 x io- 5 ] ,., y n/YTr 100 4.4 Ec/M (gt:, /1i.c )- . Thus it is seen that meson production exceeds anti-nucleon production by a factor of the order of 60 for the entire primary spectrum, but that the two are comparable for primaries of energies ~ io 11 or io 12 e.v. -75- IX. CONCLUSION. In the preceding sections ·we have used standard perturbation methods and the pseudoscalar meson theory to compute the cross-sections for the production of antiprotons in meson-nucleon and in nucleon-nucleon collisions. These were then ap- plied to determine the numbers of antiprotons to be expected due to these processes in the interaction of energetic cosmicray particles with the nucleons of the atmosphere. These numbers were found to be very small, on the order of lo- 6 (g 2 /1lc) 4 per primary particle, so that it is not too surprising, on this basis, that antiprotons have not yet been observed in cosmic radiation. However, it is of value to compare antinucleon pro- duction with meson production, as was done at the end of the last section. There tt was found that meson production exceeds antinucleon production by a large factor of about 60 for the entire primary spectrum. (Note that the meson production calcu- lated is itself too small to account for the observed numbers of mesons in cosmic rays.) But for primaries of extreme ener- gies, the cross-sections become comparable, and one would expect about as many antinucleons as mesons to be produced. Of course, there is evidence that at such energies multiple production occurs, and one is inclined to doubt that the simple perturbation treatment used here is adequate. Fermi 7 has gone to the opposite extreme and used a statistical treatment to estimate the numbers of pi-mesons and antinucleons to be expected in energetic collisions. He assumes -76- that the entire energy of the collision is concentrated in a sme.11 volume which is a sp here of ra dius flc/y but contracted in the direction of motion by a Lorentz factor 2M/ W, where W is the total energy in the center-of-mass system. This energy is assumed to be distributed among states containing different numbers of pi-mesons and nucleon pairs, according to their statistical weights. By so doing he finds reasonable numbers for the multiplicity of meson production . At low energies of about 10 to 100 Bev. for the incident nucleon, only about 0.002 times a s many a ntinucleons as mesons are produced. (The cros~ sections calculated by p erturbation methods are in a ratio of about 0.1 g 2 /'hc a t these energies.) At extreme energies, great- er than about 10 3 Bev., the number of mesons to be expected is l given as about 0.33 (E /M) 4 , with the number of a ntinucleons , as 0.76 (En/M)-4-. n Here En is the energy of the incident nucleon in the laboratory system. Both these numbers are on the same order of magnitude, in vague agreement with our results as to the cross-sections. Fermi assumes a total cross- section for 2 ~ 26 all processes of aboutrr(1'lc/p) =129(1'l.c/M) = 5.7 x 10cm 2 • This is much larger than any of the cross- sections obt a ined here, unless one takes a very large value of the coupling oonst ant. The maximum of our cros s- section for the production of antinucleons in nucleon-nucleon collisions, fr om the table on p. 66, is about 1.8 x io- 3 (fic/M) 2 (g2/1ic) 4 s.o x io- 31 = (g 2 /fic) 4 cm 2 . While both methods give comparable values for the ratio of the numbers of pi-mesons to antinucleons produced in colli- -77sions at extreme energies, Fermi's statistical treatment would seem to predict a smaller relative total production of antinucleons than the perturbation method. Since antiprotons have not yet been observed, this may indicate that his method and viewpoint are somewhat closer to reality than the conventional theory used here. Antinucleons produced in the very energetic collisions observed in photographic emulsions would have such a high energy that they would not be expected to be annihilated in the same emulsion, and they would thus be indistinguishable from ordinary nucleons. One could hope to observe antiprotons on~ lower d own in the atmosphere after they had lost most of their energy, and it seems indicated that their total number would be very much smaller than the numbers of mesons or nucleons in cosmic radiation. -78- APPENDIX I. ANNIHILATION OF ANTIPROTONS IN BOUND STATES. An antiproton in hydrogen will ultimately be captured into an S-state orbit about a proton. It is of interest to de- termine the lifetime against annihilation in such a state~ First, it can be shown that annihilation is forbidden if the state is a singlet ( 1 s 0 ). The parity of such an S-state with a nucleon and an antinucleon is odd. (This is the same as the case of positronium, cf. reference 21). The two mesons coming off can then have only odd values of total J, i.e., J ~ 1, 3, etc., since they have zero spin. Hence annihilation into t wo mesons can occur only from the triplet (3s 1 ) state, and not from the singlet state, since total angular momentum must be conserved.* It is instructive to obtain this conclusion using previous methods. Using pseudoscalar coupling, it is shown in Section II that the matrix element for the transition is pro- '\'f1'fc:=+f·'* {e:-et·1) +~ portional to , the Dirac wave-function for the initial proton and where 'f~ is f 1 that for the proton in a negative-energy state which fills the hole which represented the antiproton. ,.. q: ( f , q) is the four- momentum of one of the emitted mesons. tic limit, * In the non-relativis- tf 'fc: = 0 because of the orthogonality of states 1" In the singlet state, annihilation would be most likely into two charged mesons and a neutral meson, each taking on the average 1/3 of the available energy. -79of positive and negative energy. only on the term Hence our result depends '1'/'' ~-~ \ft or fr oc} 'fr: , if we take the 'It z-axis along the direction q of the mes on. (This is permis- sible in a singlet S-state, which has spherical symmetry and no preferred axes.) The same result would follow from the scalar coupling, in this limit. We must, however, take into account that the initial singlet state must be anti-symmetric in the sp i n parts of the wave-functions of the initial particles (proton and antiproton). To convert the matrix element into a form in which the wave-functi on of the antiproton appears, we use the charge- conjugation operator C, which has the property that if + represents a particle of positive charge, and satisfies Cf represents a particle of the Dirac equation, then ~*=: negative cha rge which is really a hole in the sea of negativeenergy states of the former positive particle2 2 . matrix element can be written down, putting Then our O(~ = 0, tf = c-• f.) "t+c* =ere-'*, ( I.Ol) M = tf * 0 tL = L: '\'f: Qoc~ ti~ 0( fl L ( <J' c_, *JQt o Pt ~ = ~ erYc~: oal P+p= L. r..JrP cr,,t~ , O{ -~ where N:: c- l"~o. ~~¥ ~y Now let superscripts (1) and (2) refer to the two spin states (up or down) of the initial particles. For the singlet state (total spin zero), we must use, instead of (I.01), ( r . 02) M ( ::: <i) l2.) cz.) ti:- ~ Nyp <J'y tp - ft tp ci>J · -80- In the usual system of CL,~ matrices, in which they are Her~ mitian, a suitable charge-conjugation operator 0 is given by c = -i (3 0(1 ( r. 0.3) Then 0 -1'"' - C, and l 00 00 01 -1 0 0 1 0 0 -1 0 0 0 J og1 oo1 go _ o] 001 N=Co<= [ } (I.04) 0 -1 Non-relativistically, (0 0 0 1). h.) :. 'f lt):. 'fll) -: . (0 0 1 0), <f l'2.) = d,, T Putting these into (I.02), we see that M =0 so that the transition is forbidden in the singlet state.* The annihilation into two charged mesons for an S-state of the bound proton and antiproton must therefore go only in the triplet state. ( r. 05) where T-I - The lifetime T is simply given by 4 v cJ ( 0) t cp ( nz.. = Tt ro c: r 4> { )J 7. 0 0 'L I {o) }'2. is the density of nucleons at the origin, and \,t.. ~ v is the relative velocity of the proton and antiproton. 6 (0) is the zero-energy limit of the annihilation cross-section, which is given by (3.06). We multiply by a factor of four since we must use the sum over initial spin states, and In the annihilation of a positron and an electron, in the ground-state of positronium, into two gamma-rays, it turns out that one obtains for M, ~{;" 'Ys ~o -~ha~ Ys- , and 'l~ g):. +.: '[ N = C-1* 0 :. C "(-s; :: 1 0 0 0 0 0 0 0 1 0 -1 0 so that M ~ 0, and the process is allowed in the singlet stateo I~ is orbidden in the triplet state from other considerations 2 • f -81- not the average, which is used in determining the cross-section. cp (r) is the wave-function of the ground-state of the proton-antiproton system, given by the usual formula ~ (r} (I. 06) where a \1fCA. ~-i e-r/a.. = 211.2c 2 /Me 2 is the Bohr radius, using the reduced mass of !M. (I. 07) - Then (I.05) becomes - 1 T =-L f£)2._f~) 8 \t:\c \l\c. 3 M 1' -82APPE NDIX II. MU-MESON DECAY SPECTRUM. As a further application of the formulas of section I for momentum-space integration, we shall work out the energyspectrum of the electrons from the decay of the mu-meson. Such spectra have been worked out by Tiomno, Wheeler, and Rau20 for the five usual beta-interactions. V!e shall use the com- pletely anti-symmetric interaction of Wigner and Critchfiel~. Then the matrix element for the interaction can be written (II.01) H =~ '1''"I • 4> f 1 iv, -t3• c;2. cp2. ~1 - 'W2. ~ <PJ <l'J I "'! '\(, ... cpIf "'4 "'" where i( is the plane-wave Dirac spinor for the decaying meson, and 4, 'I', f are spinors for the particles produced, - - which have momenta Pl, p 2 , p 3 , energies E1 , E2 , E3 , and masses m1 , m2 , m3 , respectively. M is the mass of the mu-meson. The components of the Dirac plane-wave spinors are tabulated 15 in Heitler , p. 86, and we can use them to evaluate the sum of the absolute squares of H over the spin states of the final particles. In the c.m. system, the state of the initial (resting) meson is given by "i( -: . (0 0 0 1). Then we get, us- ing also conservation of energy and momentum as embodied in ( 1. 06) ' (II.02) where 3 L.IH\"t - L IHLl2. i.= ( -83- Assuming the decay to produce an electron of mass m1 =m and two neutrinos of zero mass, we obtain L \HI2 ::. a2 ( ( E1 + m) (M2 + m2 - 2ME1 ) + ( I I.04) 2 E 2 (M2 - m2 - 21if£2) + E3 (M - m2 - 2ME 3 ) j /8E1E 2 E3 • The decay probability per second is given by the usual formula dw ( II.05) vvhere ri= is the density of final states, given according to ( 1. 20) as (II. 06) where we have already integrated over angles, which do not appear in L jHJ2 . In the integration over neutrino energiei: E 2 , E3, the last two terms of (II.04) give an equal contribu- tion. Using (1.12-.13), we obtain for this case, ( II.07) J JEzdE 2 ~ dE2 :. p 1 , tp fE22dE2 = f2 Pl [ .3(M - El)2 where p 12 -- E 1 2- m2 • 1 (M - E1), + P12 J Hence the pr opabili ty that an electrcn is emitted with an energy between E1 and E1 + dE 1 is given by l. dw =- (I I. 08) ~ f• 4 ( 2 TT )3 ~ 1 C (, + 2. EI l IV\ 'l.. - M M + W\ "- ) with E1 S' i YV\ ( M '2. - 2. 3I ,-11.A E '2..1 d EI 0 I W\ .3 I I (\I\ M + WI '2.) "- E I < EI } = (M2 t m2 )/2M, by (1.16). Neglecting m << M, this simplifies to (II.09) dw - G"l.. M -2(2n) 3 PE 1 (M -~ E1)d El , ..vi <E,<i_ M 1' c"'' 1 -84- Then the mean-life of the mu-meson, -c; , is g ot by integrat- ing over E1 : I (II.10) 1: Taking the meson ma ss a s M-::.. 210m, and "t :. 2.15 x io-6 sec., we find G:. 0.22 x io- 49 erg-cm3. - 85- APPENDIX III. (a) ADDI TIONAL PROCESSES WITH ANTI PROT ONS. Charge Exchange. A proton colliding with an anti proton could exchang e a charg ed meson with it to form a ne utron-antineutron pair : ( III.01) Pl f,J (Fig. III.l) This is represented by the diag r am of Fig. III.l, and it is seen to be analogous to proton-neutron scattering , except that one of the nucleons is moving backwards in time. In f a ct, t he cross-se c tion in lowes t order will have t he s an e for m a s that for scattering with one meson exchanged. The matri x element is given by (IIr.02) H == 4rr~i'-t '"c..,__ f(r2.) ys- t t~l.) · [ lf i I - I ) ~- f "l. ] - I r ~ ( 1 l ) . y!; t ( I )) with ( r r r. 00) 'f, t (r. ) =- M ~ ( r. ) , r2. t t; 2.} -= - rvt 4' trl. J Cf, tli,) ~ M tl1, }) 'f~+ ('ti.) =--MtC1 ~ J The cross- section is t hen (III. 04) with -86- (III.05) v :. ( 2p/E) c in the c.m. system. and Thus the differential cross-section is (I II. 06) where Q -= tem. \p - qI is the momentum transfer in the c.m. sys- For large energies the total cross- section is (III.07) ~ rr :J 'i 2.M (E 0 +M) iTj 'f 4 EL ) where E 0 is the energy of the incident antiproton in the laboratory system. At low energies, if v is the relative velo- city of the pair, (III.08) - 1r.'1" 12. M't. (~)~\X)'i r c Thus this charge-exchange reaction has small probability in comparison with annihilation of the pair- into two mesons, for the pseudoscalar coupling. In the scalar theory, however, with l's- ::. 1 in (I I I. 02), the cross-section is larger, and (III.06) becomes j '/ (III.09) (4M'l.+Q.2,)2.. ( G ~+ r-<-r- dJ1 so.that at low velocities (III.10) This is larger than the annihilation cross-section, except for energies less than about 1 e.v. Of course, if an anti- neutron is heavier than an antiproton, just as an ordinary neutron is heavier than a proton, this mass difference would -87- act to prevent the exchange of charge at low energies (less than about 1 Mev.). But on the scalar theory, an antiprotm slowing down from about 100 Mev. would be seen to disappear and reappear as it lost and recovered its negative charge in collisions. (b) Scattering of Antiprotons. In lowest order, the scattering of an antiproton with a proton would take p lace through the exchange of a neutral meson in accordance wtth the di agrams of Fig. III.2. fl. A -~-=-'b_ P(p,) ?-(p1) (Fig. III.2) lb) The notation is the same as in III(a). We can write down the matrix element immediately: (III.11) H = 4rrj: ~Lc2. r f (rl.) ys-t(;J [lr.-,,)~-r;tJ-'· 1 f(1t Jys f (r·) -fl;. )ystli~J[fy,-tf2t-r~]- f(r,)yst{r,J}. Working out the cross-section in the usual way, we obtain I (III.12) d.6" ~ jo"f lbE2. [ ( 1 _p._1- - Lt-El. )-2 + \1- ~,f'( I+~ t+ (1 +f:fJ JJl where E is the energy and Q:. l p - q I is the momentum trans- fer in the center - of-mass system. The non-relativistic cross- -88- section is a constant: ( l -~)-2. 4H,_ (III.13) The scattering of antiprotons with neutrons takes place through formation of a virtual charged meson. similar to Fig. III.2(B) . The diagram is The cross-section is just the first term of (III.12), so that the total cross-section in the c.m. system is ( III.14) ll.J ., 4E2 Non-relativistically this reduces to (III.13), replacing g 0 by g. Thus scattering is seen to be of smaller probability than annihilation at all energies, in the pseudoscalar theory. (c) Annihilation into Gamma Rays. The second diagram of Fig. III.2 above leads one to ask of the possible importance of a process in which the antiproton combines with a proton to form a virtual neutral meson, which then decays into two gamma rays as neutral mesons have been observed to do. The cross-section for such a reac- tion can be worked out in terms of the mean-life neutr a l meson, as follo ws. Pl f•) (Fig. III. .3) ~ of the -89The mean-life is given by ~rr \H 1\'L r' ~ (III.15) where H1 is the matrix element for the decay, and volume of momentum space, given by ( III.16) r is the I 4n-1'- 3 I F Z{2rrt..c.)3 2t2rrtd where q ~ 2f is the energy of each gamma ray in the c.m. sysl tem. On the other hand, the annihilation cross-section is 2rr ~v (III.17) l't. I H o..v F) = 2p/E is the relative velocity in the c.m. system, ~ith Pl = (E, p), B2 : (E, -p) the four-momenta of the inciwhere v/c dent proton and antiproton. (III.18) and f Then also _ 4rrE2. - 2 ( 2TT 1' C ) 3 IHI a2v is the average over spins of the square of the matrix-element, given by ( III.19) H = .f4rr jo ~c. f lp:i. l 'Vs- +(r,) · l l r. + ri.) ,_ - r J_, v-zt: H, , L where the factor -{2f. corrects for a factor included in H1 because the initial meson was there free. i ng Thus we g et, us- ( I I I. 15) , (I I I. 20) t: ) r : (;T) Non-relativistically, with E-:. M, p-=- tMv/c, (III.21) The factor (f ~ W\ (1'~ h/]1't - lo- 8 for -r....., io- 15 sec. Thus this process -90- would have a cross-section even smaller than the direct annihilation into two photons, the cross-section for which is g iven by 15 (III .22) -91REFERENCES . 1. P.A. M. Dira c, Proc. Roy. Soc. All7, 610 (1928). 2. P.A. lvI. Dir a c, Proc . Roy. Soc. Al26_, 360 (18.30) . .3. o. Klein, Ark. f. Mat. Ast. och Fys . .31A, #14 (1945). 4. N. Arley, Kgl. Dansk . Vid. Selsk. 2.3, # 7 {1945). 5. J. McConnell, Proc. Roy. Ir. Acad. 50, 189 (1945) and 51, 17.3 (1947). 6. H.A. 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Phys. 21, 144 (1949 ). 24. C.L. Critchfield, Phys. Rev. 63, 417 (1943). -93- I I / I / v 0 0 0 co I ~ ......... \!:: w 0 (\j Cf) + w ~ I '.o.. r0 <.9 LL \ ~ t 0 "'" ~ 0 r0 + ~ I- + -0 0 z C\I - z w Cl z ~ 0 w z .Q_ ¢ \ 0 LO >(!) a:: w ~ z w ~ 0 ~ · - - 1 - - - - - - -,_____ a- 01 xi( co ~ ........,,,, ~lL} £( ~l )/.JJ ~ Q ro ¢ (.\.) 0 0 0 ~