production and annihilation of negative protons: iii

Production and Annihilation of Negative Protons: IIIAuthor(s): James McConnellSource: Proceedings of the Royal Irish Academy. Section A: Mathematical and PhysicalSciences, Vol. 56 (1953/1954), pp. 45-65Published by: Royal Irish AcademyStable URL: http://www.jstor.org/stable/20488562 .

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I 45 1

5.

PRODUCTION AND ANNIHILATION OF NEGATIVE

PROTONS-III.

By REV. JAMES McCONNELL,

St. Patrick's College, Maynooth.

[Read 25 JANUARY. Published 20 JULY, 19,54.]

CONTENTS. PAGE

INTRODUCTION AND SUMMARY. 45

PART I.

PRODUCTION OF NUCLEON PATIRS IN THE PSEUDOVECTOR COUPLING 'THEORY.

1. The meson spectrum of a moving nucleon ... ... ... ... 47

2. Production of nucleon pairs by meson-nucleon collisions in perturbation theory ... 48

3. Production of nucleon pairs by proton-neutron collisions in perturbation theory ... 50

4. Production of nucleon pairs by meson-nucleon collisions in damping theory ... 51

5. Production of nucleon pairs by proton-neutron collisions in damping theory ... 53

PART II.

PRODUCTION OF NUCLEON PAIRS IN TrE PURE PSEUDOSCALAR THEORY.

1. The mneson spectrum of a moving nucleon ... ... ... ... 54

2. Production of nucleon pairs by meson-meson collisions ... ... .. 55

3. Production of nucleon pairs by meson-nucleon collisions ... ... ... 57

4. Production of nucleon pairs by proton-neutron collisions ... ... ... 59

PART III.

ANNIHILATION OF NUCLEON PAIRS.

1. Annihilation into mesons in the pseudovector coupling theory ... .. 63

2. Annihilation into mesons in the pure pseudoscalar theory ... ... 64

PART IV.

DISCUSSION OF A HIGH.ENERGY NUCLEON-NPUCLEON COLLISION. 65

INTRODUCTION AND SUMMARY.

SEVERAL theoretical studies of the negative proton have recently been

made with a view to its possible detection. Helstrom 1 has investigated

the production of nucleon pairs by negative meson-proton and by nueleon

nucleon collisions, and the annihilation of proton pain into mesons. A

fuller discussion of the annihilation process has been given by Ashkin,

1 C. W. Helstrom, Phys. Rev., 78, 88 (A), 1950.

PROC, R.I.A. VOL. 56, SECT, A, [9]



46 Proceedings of the Royal Irish Academy.

Auerbach and Marshak.2 These calculations were performed by the perturbation methods commonly employed for electromagnetic processes. On the other hand Fermi " made a statistical approach to the problems

of nucleon pair production and other high-energy events, and it has been

claimed that an experimental verification of his results comes from a

nucleon-nucleon collision observed by Lord, Fainberg and Schein.4 It therefore seems desirable to resume the earlier calculations on the

production of negative protons by meson-nucleon collisions5 and to extend them to the case of inelastic proton-neutron collisions. In I we dealt only with pseudoscalar mesons having pseudovector coupling with the nucleon, while in II we considered also vector mesons. It now

appears that the nuclear force r-mesons are pseudoscalarj but the nature

of their interaction with nucleons is still undecided. We thus consider only pseudoscalar mesons, but allow the possibilities of either pseudo scalar or pseudovector coupling. In order to make a comparison with the results of other investigations we employ both perturbation methods and radiation damping theory.

The calculations are simplified by adopting the Weizsiacker-Williams method to resolve the field of a fast nucleon into a field of pseudoscalar

mesons. This method is approximate and no great numerical accuracy can be claimed for our results, especially in the case of proton-neutron

collisions since they involve two resolutions of the field of the particles. For pseudovector coupling perturbation theory gives cross-sections which diverge at high energies, while damping theory yields cross-sections which

have maximum values about 10-28 cm.2 and tend to zero with increasing

energy. The case of pseudosealar coupling is restricted to a charged meson

theory on account of a recent study of the applicability of the Weizsiacker

Williams method to meson fields.7 Perturbation theory gives results

which are reasonable at all energies and do not differ much from those

derived by damping theory, whenever a comparison can easily be made.

A more general solution of the integral equations of radiation damping

theory would be very difficult and we treat the problem by the pertur

bation method only. The cross-section for pair production by the

collision of a fast meson with a nucleon at-rest rises from zero when the

meson energy is about 4 Bev to a broad maximum attained when the

energy is about 30 Bev, and then decreases slowly to zero. The cross

section being proportional to the sixth power of the pseudoscalar coupling

?

2J. Ashkin, T. Auerbach and B. Marshals, Phys. Rev., 79, 266 (1950). 3E. Fermi, Prog. Theor. Phys., 5, 570 (T950). * J. J. Lord, J. Fainberg and M. Schein, Phys. Rev., 80, 970 (1950). 5J. McConnell, Proc. R.I.A., A, 50, 189 (1945); ibid., A, 51, 173 (1947)?to be

quoted as I and II respectively. 9 Cf. J. Steinberger and A. S. Bishop, Phys. Rev., 86, 171 (1952).

7 J. McConnell, Proc. R.I.A., A, 55, 183 (1953).



McCoNNiwLr-Production and Annihilation of Negative Protons-Ill. 47

constant g depends very sensitively on its value, and, if we take = 4, he the maximum cross-section is 5 x 10-27 cm.2. The pair production cross

section when a proton and neutron collide has a threshold energy of the incoming particle about 7 Bev and is proportional to g9. It starts from

zero, has a broad maximum value 9 x 10-27 cm.2 reached at an energy

200 Bev approximately, and then falls slowly down to zero.

The annihilation of nucleon pairs into pseudoscalar mesons is investi gated for both pseudovector and pseudoscalar couplings. The mean life time in lead of a slow negative proton is 2 x 10-11 sec. approximately for

pseudovector coupling and about one hundred times greater for pseudo scalar coupling. In either case it is more likely that mesons rather than

photons will result from the annihilation. A comparison with Fermi's theory shows that our results differ very

much from his at the high energy occurring in the nucleon-nucleon collision detected by Lord, Fainberg and Schein. According to our

calculations the probability of pair production at that energy is very

small.

PART I.

PRODUCTION OF NUCLEON PAIRS IN THE PSEUDOVECTOR COUPLING THEORY.

1. The meson spectrum of a moving nucleon.

The investigations of I on the production and annihilation of negative protons assumed that the nuclear force mesons are pseudoscalar and that

they have pseudovector coupling with the nucleon. The production problem for meson-nucleon collisions was reduced to that for meson-meson

collisions by resolving the field of a moving nucleon into a field of virtual

mesons. This resolution was performed by Heitler and Peng,8 who adapted a method for electromagnetic processes due to v. WeizsAcker 9 and Williams.10

The calculations of Heitler and Peng have been re-examined and a study has been made of the dependence of the spectrum of virtual mesons on bmi, the minimum value of the impact parameter for the collisions.1' To obtain

a meson production cross-section which agrees with experimental results a

value of the pseudovector coupling constant f such that J- - together

h h eMs with bmn equal to or 4 - , M being the rest mass of the proton, is

Mc Mc

8 W. Heitler and H. W. Peng, Proc. R.I.A., A, 49, 101 (1943). ? C. F. von Weizs?cker, Zeit. f. Phys., 88, 612 (1934).

w E. J. Williams, Kgl. Dan. Vid. SeUk., 18, 4 (1935). il J. McConnell, Proc. R.I.A., A, 55, 101 (1953).

[9*]




admissible. For convenience we choose units such that h c = = 1,

Ht being the rest mass of the nuclear force meson which we take to be the

r-meson, so that M = 6-6. Then the number of positively (negatively)

charged virtual mesons with energy in the range (E, e + de) accompanying

a proton (neutron) with energy E is approximately

qpv (c) de = Nfl de (1, 1) nTE

where f 60 for bmin. = i

N -Mc

25 for b = h

In a charge symmetrical meson theory a spectrum of neutral mesons will

also accompany the moving nucleon, but it will appear that these do not

influence the results.

The subsequent calculations differ from those of I in that we adopt

equation (1, 1), that we study proton-neutron collisions and that we examine

the pair production problem also by perturbation methods.

2. Production of nucleon pairs by meson-nucleon colli8ions in perturbation theory.

We investigated in I the production of a nucleon pair by the collision

of two oppositely charged mesons. Working in the centre-of-mass system, we denote by k and e the momentum and energy of the positively charged

meson and by p the momentum of the produced nucleon, and we distinguish

between the non-relativistic (N.R.) and extreme relativistic (E.R.) cases

according as p <( M or p )) M. The matrix elements HP X, HN Ie for the

production of proton and neutron pairs respectively are

HPI = _ f 2(up* (ak) up*), N. M NR

HNJe 4-M- (uN* (ak) uN*) M (1, 2)

HPIe - 2- (UP* (ak) up*), E. E

HN]e (- -2(*N (ak) UN4

The u's are spinors referring to the nuclear particles and the components of

a are the Dirac matrices. Proton and neutron are denoted by P and N,

and P*, N* denote the proton and neutron in the negative energy states

corresponding to the anti-particles. The matrix elements HP I f, , HN Ceo for



McCoNNELLi-Production and Annthilation of Negative Protons III. 49

production of nucleon pairs from uncharged mesons vanish in the f2-approxi mation.

In calculating the differential cross-section for pair production do from

the matrix elements (1, 2) we sum over the spin states of the nucleons with

both positive and negative energy. We then obtain by the usual procedure

do 2Mf4 p dg, N.R. (1 3) d - = sin 20 f4E2 d E.R.

0 being the angle between k an-d p and dt the element of solid angle. On

bringing one of the mesons to rest we obtain

do = 4V/2 r Mf4 dE (2M2 < E'< 4M2) (1, 4)

do = 47rf4(S - 7) dEP" (El > 4M2) (1, 5)

where c' is the energy of the meson which is not brought to rest and E' the

energy of the produced nucleon. The ranges of values of E' for which the two expressions are adopted have been discussed in I.

Pair production by a meson-nucleon collision occurs through one of the

transitions Y_ +P P'+ p- + N,

Y - + P -N + N- + N',

Y+ + N- P + P + P',

Y+ + N = N' + N- + P,

where Y+, Y-, P-, N- signify respectively a positive meson, negative meson, negative proton and anti-neutron. To obtain the cross-section arising from

the collision of a meson having energy co with a nucleon at rest we make a

Lorentz transformation so that the meson comes to rest. The total cross

section is invariant for such a transformation. The nucleon then acquires an energy co M and the energy c' of the virtual mesons of its spectrum ranges from a value rather greater than the meson rest energy 1 to a limit some

what below c0 M which we take to be i co M. Pair production does not

occur unless i co M exceeds 2M2, that is, unless Eo is greater than 4M. When

c0 < 8M, we deduce from (1, 1) and (1, 4) the cross-section for pair production

4A/2 Nf6 0 ? e' i/E'(e'-22)

(co) - I

I _t -de' dE' Eo J 2M2 ,Je' Ve'(e'-2M2)

- 4NM312f6 (EO 4M)3/2 (4M < Eo < 8M) (1, 6) 3 Eo

The limits of integration of E' are considered in I.




When c0 exceeds 8M, the virtual mesons with energy less than 4M2 give a contribution to the cross-section

32 N M3f6

3 e0

while by equations (1, 1) and (1, 5) those with higher energy give

4Nf6 zijoM 'E' {El E'2

EOM J 4M2 JjM2 \ E E

_ 1 EM (EO2 - 64 M2), approximately.

The total cross-section is therefore

#(co) = NM ( 64M2 - (E' > 8M) (1, 7) 12 k q

For high energies the cross-section diverges with c0.

3. Production of nucleon pairs by proton-neutron collisions in perturbation

theory.

The production of nucleon pairs by proton-neutron collisions occurs

through the transitions

P + N P' + P~ + P + N",

P + N N' + N + P'J + N'-,

and the cross-section is obtained by a further application of the Weizsicker

Williams method. Since each of the nucleons is moving with reference to

the other, each will have a meson spectrum. We are not interested in the

energies of the produced particles and it will therefore be sufficient to consider

the spectrum of one of the original nucleons, the other being at rest, and

multiply the cross-section by 2. Neutral mesons had no influence in meson

nucleon collisions and they may now be neglected even when a charge

symmetrical meson theory is adopted. The cross-section for pair production

by the collision of a proton (neutron) with energy E and a neutron (proton) at rest is

IF a (E) 2 |4 (co) qv. (Eo) dEo, (1, 8)

4M

from which it appears that the process does not occur unless B exceeds

8M, that is, about 7 Bev.



MICCoNNELL--Production and Annihtlation of Negative Protons-IlI. 51

When E lies between 8M and 16 M we use equation (1, 6) and obtain

a (E) = 8N2M3/2f8 ifB (Eo - 4M)3/2 dE 3tzrE 4M Eo

= 128N2M3Pf ( fE- 8M\312 E -SM\

37rE k 8M \ 8M )

+ 2 tan [(E-SM) (E -SM)' }

(8M < E < 16M)

When E is greater than 16M, we have approximately

8N2MI2p SJM (Eo - 4M )3/2 N2Mf8 IE { 64M2 ) a(E) = I - dE,0 + I Eo+ jdc0

3-ffE J4M coJITE M ~ Co

NMf8IE 32 M2 F N2fs( + 32 M_2 log (E > 16M). (1, 9)

The cross-section is proportional to E for large energies.

4. Production of nucleon pairs by meson-nucleon collisions in damping theory.

To avoid the diverging cross-sections we employ the radiationi damping

theory of Heitler and Peng.12 This amounts to solving a set of integral

equations. UBA = HBA + irrrXpc HBC UCA (1, 10)

and replacing the matrix element H by U in the formula for the differential

cross-section. Since Hp I ( and HN I 0 vanish, it may be deduced that Up I 0 and UN v 0 vanish so that neutrettos do not influence the pair production of

nucleons. We transcribe from 1 13 the differential cross-section for nucleon pair

production by the collision of a meson with energy E' and a meson at rest,

E' being the energy of the produced nucleon,

4/2-rMf4 dEr d= 2 (2M2 cE' < 4M2)

(1 + a> M

do _ 129G(/r6 - F'2) dE' (E > 4M2) f,4

i2 W. Heitler and H. W. Peng, Proc. Camb. Phil. Soc, 38, 296 (1942). 13 Cf. equations (2, 9) and (2, 11) of I.




where 4

a = 4M3 f4. 3

Proceeding as before to calculate the cross-section when a meson with

energy c0 collides with a nucleon at rest, we obtain

E' + i 'V 2M2) dE'

qS (ED) - 4V2Nf~~~~~~~ jcqM~ JE' -

j Vf' (f'-

27M2 __ __ __ __ __ )

32NN6 { / 1' M

a 32fo a EOM - 3 2 2log (1 + aVj/ EaM

-

+ >/ aVi EoM M2 M

1 + a/IEOM M2)

(4M < co < 8M) (1, 11)

Though the cross-section depends on the coupling constant in a complicated

way, numerical computation shows that it is roughily proportional to f-2.

When E0 > 8M, the contribution from the range of E' between 232 and

4M2 is very nearly

18N MSf2.E'

and adding to this

1296N FcoEm IE ' EEl d,E' idE'

_Mf2'EO - 4M2 d iM2 \e5 E 6

we obtain approximately

E (eq) 2 1 0-27jYj}. (Eo > 8M) (1, 12) M6f2,E0 I

In forming an estimate of the cross-section we recall that it is expressed

in units of that is, 1-92 x

10-26 cm.2. Let us choose b n. so in us

[LCmii.-

0

that N = 25. Pair production does not occur unless the energy c0 of the

meson is at least 4 Bev. The cross-section rises quickly to its maximum value

5-2 x 10-29 cm.2 attained when E0 is 7 Bev, and then slowly tends to zero

with oj1. The variation of the cross-section with energy does not differ very

much from that derived in I except that there, on account of the different

form of the meson spectrum of the moving nucleon, the cross-section tended

to zero with Ej II



MCCoNNuLL-Production and Annihilation of Negative Protons-Ill. 53

5. Production of nucleon pairs by proton-neutron colliisons in damping theory.

The cross-section a (E) for pair production of nucleons by a proton neutron collision, the energy of the incoming particle being E, is given by

(1, 8) when A (ED) is taken from (1, 11) or (1, 12). When E is less than 16M, we have approximately

=36N2 [lB dE0o 2' aE = M6E J ? A VJIEOM Ma log(1 + aIoMM )+-J

= 36N2YM IE M - 2M tan-1E_8M 7TMBE 8M- 8M

3 E-SMp,log (1 + aM ESM'); 2a 8M S 18M

(8M < E < 16M) (1, 13) For larger values of K

a (E) = 36N JM de Co M M2 - log (1 + akE oM - M2) +4 rrM6E JM ? a a,

+ 5ON2 i BE dE - 0 0278MA2

rM5E 8M -o (CoM

= 3MNE 2M - ZM - !log (1 + aM) - irM6EI 2 2a

+ 5ON2 {log 16 - O 135[ ( E ) ]}

(E > 16M) (1, 14)

The asymptotic value of the cross-section as E increases indefinitely is

5ON2 E log -, rrM5E 16M

which tends slowly to zero. We note that it is independent of the coupling constant.

A numerical evaluation of (1, 13) and (1, 14) for different energies shows

that a (E) has a broad maximum reached when E is about 30 Bev. For

bmin - M the maximum cross-section is about 6W2 x 10-29 cm.2, which is Mc

roughly the same as the maximum value in the case of a meson-nucleon

collision. These maximum cross-sections are about 1,000 times smaller than the experimental cross-section for meson production by a nucleon-nucleon

collision when the energy of the incoming particle is 3 Bev or more.




PART II.

PRODUCTION OF NUCLEON PAIRS IN THE PURE PSEUDOSCALAR THEORY.

1. The meson spectrum of a moving nucleon.

The energies involved in a collision which causes the production of a

proton or neutron pair are very much greater than the rest energy of the

meson. The energy dependence of the matrix element for the emission or

absorption of a pseudoscalar meson by a nucleon is of a higher order when

the coupling is pseudovector than when it is pseudoscalar. For this reason

the case of pseudoscalar coupling was not examined in the calculations of I.

However, the argument loses much of its weight, if the pseudoscalar coupling

constant g is appreciably greater than the pseudovector coupling constantf

Moreover, the recently developed renormalisation procedure of quantum

electrodynamics has given pseudoscalar coupling new theoretical importance.'4

So it is worth while to study the nucleon pair production problem with a

theory of nuclear forces due to pseudoscalar mesons which have pseudoscalar

coupling with the nucleon, that is, with a pure pseudoscalar theory.

The pure pseudoscalar field of a fast moving nucleon has been resolved

into a field of virtual mesons for different minimum values of the impact

parameter.15 Taking as before h l c = ju = 1, we find that the number

of the positive (negative) mesons with energy in the range (E, c + dE) which

accompany a proton (neutron) with energy E is approximately

qp8. (E) d de 2 D,8, (2, 1)

where A EM for ?M < 1 I E

(2, 2)

B EM + C for - > 1 E E

and A, B, C are constants depending on bmin . An application to the problem

of meson production by a high-energy proton-neutron collision in a charged meson theory 16 yields results, which are in qualitative agreement with those

obtained by direct calculation and which suggest that

bmin. h g2 Mc' b

14 Cf. K. M. Watson and J. V. Lepore, Phys. Rev., 76, 1157 (1949); E. Corinaldesi and G. Field, Phil. Mag., 40, 1159 (1949), ibid., 41, 364 (1950).

15 J. McConnell, Proc. R.I.A., A, 55, page 169 and corrigenda (1953). The values of A, B, C given in the table on page 172 should be reduced by approximately a factor 2 or 3.

i? J. McConnell, Proc. R.I.A., A, 55, 183 (1953).



MCCONNELL-Production and Annihilation of Negative Protons-III. 55

This value of bmin. gives

A = 0-35, B = 0, C = 045.

U2 We accept the above values of bmin and h and for convenience make

the approximation A = 0 - -4, B = . (2, 3)

If neutral mesons are included in the Weizsiicker-Williams spectrum,

the results for meson production do not agree with those obtained by direct

calculation. We shall therefore disregard neutrettos in our investigations.

This fortunately reduces the complexity of the mathematical problems

which arise.

2. Production of nucleon pairs by meson-meson collisions.

The first problem to be considered is the production of a nucleon pair

by the collision of two oppositely charged mesons. For definiteness let us

examine the production of a proton and a negative proton. We work in the

centre-of-mass system, p being the momentum of the proton and k, c the

momentum and energy of the positive meson. Then E must be at least equal

to M, the rest energy of the proton, and k may be approximated by E. The

pair production occurs through the transitions

Y-+ P* *-N, N + Y+ >-P,

where P* denotes the particle in the negative energy state corresponding

to the negative proton, and the matrix elements for these transitions are

respectively

- J-tig (U*N P2 UP*); j _

i (u p P2 UN),

P2 being a Dirac matrix. The compound matrix element is then

=27rg2 (U* p (ak) UP*)(24 Hp I e = 7r s1(k 1 (2, 4)

Similarly the matrix element for the production of neutron pairs is

HN I- 2wg2 (u*N (t&k) UN) (2, 5) E M2 ]+jp +kj12'(2)

p being the momentum of the neutron.

We may proceed to a solution of the integral equations (1, 10) by the

method adopted in I. However, the presence of M2 + I p ? k j 2 in the

denominators of the compound matrix elements leads to mathematical



56 Proceedings of the Royal Irtish Academy.

difficulties, of which there is no ready solution. If we make an approximation

in which quantities of the order -P are neglected we obtain

M3

and the integral equations have the solution

7T2

UPI = M (u*p (ak) up.).

1 + up +12M3

Since M = 6-6 and g2 - 4, the influence of damping is small in this case

In the general case perturbation theory yields cross-sections which do not

diverge for high energies, so we have not the same compelling reason for

employing the radiation damping theory as we had when dealing with

pseudovector coupling. An inspection of the different terms of the integral equations suggests that the energy dependence of Up I, is the same as that of

Hp I eat high energies. We therefore adopt perturbation theory in the hope

that damping effects are not too strong.

Equation (2, 4) gives the differential cross-section

o= g4 M2 + p2(l - co-s2b) d 2E {M2 + p2 + k 2- 2pk COSG}2

where 9 is the angle between p and k. In the approximation p (M H we obtain

the total cross-section

3 -P4 N.R. r 2M3

and when p )> M

+4 +1 M2+ p2(1-v2)

f_ {M2 + 2p2 (1-tt)}2

=ffr4 [M + 2p2l M2 + 4p2 4 (M2 + 4p2)] 4p-2 2 log

M2 M2 + 4p2 i

that is approximately,

- ff(Qog - i). E.R.

Finally we transform to the frame of reference in which one meson is at

rest and denote by E' the energy of the moving meson. Then we find as

in I that

P2 =2 2



McCoNNvELm-Production and Annihilation of Negative Protons-Ill. 57

and the approximate values of # yield the cross-sections

i-q4 0.1 (c ) =2M3 2 - M12 (2M2 < E' < 4M2) (2, 6)

(1g4 2 E' 0 f'') log ( a ) > 4M2) (2, 7)

The- same total cross-sections for the production of neutron pairs may be

derived from (2, 5).

3. Production of nucleon pairs by meson-nucleon collisions.

The investigation of pair production by meson-nucleon collisions involves

a complication which is not present for pseudovector coupling, viz., the

meson spectrum of the moving nucleon assumes a different form in different

ranges of energy of the virtual mesons. When a meson having energy E0

collides with a nucleon at rest, we transform to the Lorentz system where

the meson is at rest and the nucleon has energy e0 M. According to (2, 1),

(2, 2) and (2, 3) the spectrum of virtual mesons accompanying the moving

nucleon is

qps. (e') de' = q' (E') dE' = de', for E' < so, (2, 8)

qp8. (E') de' "

q" (e') JE' = Ag2 de' for E' >Eo* (2, 9) IT 4E

The total cross-section is then

tcEo M P

(Eo) = I 4(E) qp8 (et) de', (2, 10)

U 2M2

where 4 (e') denotes 4' (c') or 02 (E') as given by (2, 6) and (2, 7) for different

values of E'. To facilitate the calculations we distinguish the following

cases:

(a) 4M < so < 8M; (b) 8M < co < 2M2;

(c) 2M2 < Eo < 4M2; (d) 4M2 < C0,

calling the respective cross-sections a (E0), A (E0), cD, (), d (E0).

In case (a) the value of 1 c0 3 does not exceed 4M2 and we take 0q (E') in (2, 10). Besides E' > 2M2 > 8M > qo, so (2, 9) is adopted and hence

'Pa (E ) = t; 1 (a') q" (a') dE' s 2M2

Ag 6 jeoM Ic' da'

2M3 2M2 2 C

Ag co - 4M _ tan- - 4




In case (b) C' > co and both (2, 6) and (2, 7) are employed

f4M2 d fiEoM 'pb I(C) = 1' (,') q' (CE) d + '02 (C') qi (C') de'

, 2M2 3 4M2

-_ A91 + i log8 _ _ 2M log?fl. 312 4 CO ff1

Since C' may be less than c0 in case (c), both (2, 8) and (2, 9) must be used and therefore

(Pc (cO) = | 'A1 (eE') q' (e') dE' + 4M2 (e') q" (e') de' + OM0 (C') q" (') de' 2.M2 ic i4M2

-, /2qA (c -2M2)3I2 96A i o 2Mg C ($313 Co ~ +6A f1lpg8<>I2M- lgCo 6M3 Eo M2 1 4 Co M

Eo - 2M2 + 1tan Eo-2M2i 2M2 V 2M2$

Finally

4AM2 , , [Eo

4'd (CO) = J2M2 (C') q' (C') de' + 02 (e') q' (e') de' + l 02 (Eo ') q"(e')de' 2M2 4M2 C O

=_A9_; log ) - log M loS + - - 2 (log

8)a. CO Ag6

(og331

As co increases indefinitely, the cross-section has the asymptotic value

(log 2j:)2 Ag 6 ( _?M2)

2 Co and so remains finite.

The pair production cross-section being proportional to y6 depends very much on the value taken for the coupling constant. By considering different

energies of the incident meson we find that the cross-section starts from

zero for co equal to 4 Bev, rises to a broad maximum which occurs when c0 is about 30 Bev, and then decreases slowly to zero. On taking g2 = 4,

A = 0-4 the maximum cross-section is 0-26 h-c)2 which is about one

fifteenth the meson production cross-section for a nucleon-nucleon collision

with energy greater than 3 Bev.

By a direct calculation Helstrom 17 found that the cross-section for

proton pair production by a negative meson-proton collision attains a

maximum value about 0-03 g2 (M) when the energy of the incident meson

17 loe. cit..



McCoNNEma-Production and Annihilation of Negative Protons-Ill. 59

is 27M, and then decreases. We may put 27M equal to 4M2, and for this energy our cross-section is

0-44 Ag6 0-18 g2 92\2 - M2 .,m)

Thus while our maximum occurs at approximately the same energy, its value is six times that given by Heistrom.

4. Production of nucleon pairs by proton-neutron collisiow.

The total cross-section for nucleon pair production by the collision of a

proton (neutron) with energy E and a neutron (proton) at rest is

a (E) = 2 JE (E) qP,, (eu) d,,o, (2,11)

where 0- (E0) denotes the cross-section just obtained for meson-nucleon collisions and

qps (co) deo = q. (Eo) deo = -E-de, for E. < E wrE

qw, (Eo) deo = q0 (ED) deo = d

? for co > E

7 Eo M

For future reference it is necessary to evaluate some indefinite integrals.

I (EO) - 0s (fo) dEo

Ag6s 8M (o - 4M 82 - 4 -M _ ED an 4M iT2 V 3 4M 4 4M 4 4M

do h

EDa (,ED) ED

- Ag6 i log ED, approximately. 3f2 8

Ib 4ED) = { b (co) dLo

M2Aik (I + E log 8 - 7- Co M log_

J (ED) b (Eo) dLo

( + jlog 8 - log co + A logf9 + 2 M2 1, --)4gED CE M ED




I, (Eo) f Oc (Eo) deo

- M: t (1 + Ilog 8 - j)EO - M(logM) -?9 ( 2211)

l0M2JEo 2M2 + (E + ? 2)Mtana qIE 2M?

3 2Mk2 3 2JI 2

Je (Eo) 4' 'Pc (E)o Co

- ] 4 + ilog8- 8

lo0 +

logA +2M

2112 I,\ 812 C0 M 11 Co -22

+ 1 1 + M ' _tanl |?} + ~ / __CEo2312

Ij (ca) F |Pd (CO) d + jt

- i?[}!{>og2ED) -3M (2og j) + Ej+ log 8

- (log 8)Y]M112 ogeo.

id (CO) 'pd ('E)

Ag 1 M2 (log 2E) M2 log 2Eo 2M l Cg co

ui~~ 2~lo 31; -l g + -log~ _B, 2

2EO M ,I

Co 312 co 31

M32 F5 + logS - j(logS)2 - 2.

Eo -3

To derive the pair production cross-section from (2, 11) we distinguiish

between six ranges of values of the energy E.

(i) 8M ? E < 16M

jE

a (E) - 2 | (Pa (co) qp (co) deo

- 2Ag2 {Ja (JiE) - Ja (4M)}).

IT

(ii) 16M < E < 4M2

SM rIB

a (E) - 2 OPa (Co) qp (co) dEo + 2 0 (co) qg (co) deo 34M JSM

- 2A-2 {Ja (SM) -

J (4M) + Jb (jE) - Jb (8M)}. 7T



McCoNNEuL-Production and Annihilation of Negative Protons-Ill. 61

(iii) 4312 < E < SM2

E m 8AI

a (E) = 2 jPka (Eo) q. (ED) dEo + 2 J Oa (Eo) qf (co) deo SM E

S2M^2 . Ei T 2 J b (co) qpi (,o) dco. + 2 JM0, (,Eo) qs (co) do,

= 2AM1g2lZ E - 'a (431) + 2A9 (a (8M) - Ja (

+ Jb (2M2) - Jb (SM) + J, (AE) - J4 (2M2)

(iv) 8312 < E < 2M3E

a (E) = 2 O M0a (Eg) qa (co) dEo + 2 06 (Eo) qa (co) dEo + 4M .8M

(2M2 4M2

+ 2 JR b (co) q8 (ED) dco + 2 321 (2o) qA (Eo) deo

it~~~~~~~i + 2 4 d (cEo) qp (co) deo

=2SAMg2 Ila (8M) - Ia (4M) + b ) -b (8M)

+ 2A Jb (2M2) - J(Z) + Jo (4M2) -JC (2M2)

+ Jd (4E) Jd (4M2)

(v) 2M3 KE < 4M3

r8M F2M2 a (E) 2 J i2 (Eo) qa (co) d0o + 2 b (ED) q. (co) deo

4M SM

E

+ 2 M2 'c (co) qa (Eo) dE0 + 2 31c (co) qO (Eo) dEo

+ 2 J0 d (ED) qp (Eo) deo

=_ 2AMg2{Ia (8 ') Ia (4M) + Ib (23M2) - Ib (8M)

ITE

+ IC - I4 (2M2)}

+ 2g I C (432) JC (-) + Jd (jE) - Jd (4M2)4

PROC. R.I.A., VO.i, 56, SECT. A. (10]




(vi) 4M3 < E

f8M 2M12

(E) - 2 f MOa (o) qa (6o) dEo + 2 'b (M ) q. (co) deo +

+ 2 JOc (Co) qa (60) dc? + 2 i d (60) q0 (6o) dEo J2M-2 *4M2

+ 2 J0d (6o) qp (6E) deo B

2AM- 2

Ga (8M) -

Ia (4M) + Ib (2M2) I (8M'

WiE ~ ' )- lb (8)

+ IC (4M2) - I(2M ) Id (4M2)

2A g2 ( + Jd (jE)

- Jd (\Mj(2 12)

Numerical calculation shows that the graph of the cross-section as a

function of the energy of the incoming nucleon has a broad maximum whose

value 0-45 (A) is attained when E is about 200 Bev. When E >> 4M3,

a(E) 2AMg+ Jd (IE) - Jd

A2Mg8( 2E 3 3rE \ M3/

which tends slowly to zero as E increases indefinitely. By the combination

of one application of the Weizsiicker-Wilhiams method and the direct method

employed for negative meson-proton collisions Helstrom 18 obtained a cross

section 0-01 g4 (M) for an incident nucleon energy in the region of 30 Bev.

Taking 4M2 as an expression for this energy we obtain the cross-section

a(4 2) 042 Ya -0 02 g4 (M2

which is twice Helstrom's cross-section.

18 loe. cit. and private communication.



MCCONNELL-Production and Annihilation of Negative Protons-Ill. 63

PART III.

ANNIHILATION OF NUCLEON PAIRS.

1. Annihilation into mesons in the pseudovector coupling theory.

An anti-nucleon oni colliding with a nucleon at rest may be annihilated

with emission of two photons or of two mesons. The cross-section for

annihilation into photons is 19

e( )2 y1 y2 4y + 1 2log (y + 2 1) r 3

where the total energy of the incoming particle is y Mc2. From this it may be deduced that the mean life-time in lead of a slow negative proton is

1*7 x 10-4 sec.. The matrix elements for the annihilation of a nucleon pair into mesons

in the centre-of-mass system are the Hermitian conjugates of those in (1, 2).

The velocity of the incoming particles is p in the N.R. and unity in the M

E.R. case, and we now average over the spin states of the proton in the

two energy states. On referring to (1, 3) we see that the perturbation method gives the total cross-sections for annihilation

27rM3f4 N.R.

p

ff E2 E.R. 3

In the laboratory system where one hea-vy particle is at rest we denote by

p' the momentum and by E' the energy of the moving particle. The total

cross-sections then become

p1r M3fo N.R. p,

7hMf4 E', E.R. 6

p' = 2M being the dividing line between the N.R. and E.R. cases. The solution of the radiation damping equations (1, 10) for the annihila

tion of a proton pair into pseudoscalar mesons with pseudovector coupling was performed in I and yields

44f2

U 1 ?M (uip% (ak) up), N.R. U,Ip _ 4.

1 + -Mf 4p 3'

19 Cf. W. Heitler, Quantum Theory of Radiation, 3rd ed., p. 270 (Oxford University Press, 1954).




U61 p P- - ?La 5 (U*p. (ak) up), E.R. f 2d5

These lead to the cross-sections in the centre-of-mass system

27rM3f , N.R.

r(l + _M3f4p)2 3

277r E.R. 4f 4e6'

and 47fM3f4 N.R.

P' (1 + 3M3f4p')2

5 4r,- E.R.

M3f 4 E13

in the laboratory system.20 The cross-sections for the annihilation of a neutron

pair have the same values.

We see that for a small velocity v of the anti-nucleon the N.R. cross

sections diverge with v-1, as was pointed out by Ashkin, Auerbach and

Marshak.21 This leads to a finite life-time for a slow anti-nucleon. The

life-time of a slow negative proton in lead before it is annihilated into mesons

with pseudovector coupling is 1-9 x 10-11 sec., and it is therefore much more

likely that annihilation will occur in this way than into light quanta.

2. Annihilation into mesons in the pure pseudoscalar theory.

The matrix elements for annihilation of a nucleon pair into two mesons

in the pseudoscalar coupling theory are for the centre-of-mass system the

Hermitian conjugates of those given by equations (2, 4) and (2, 5). From

these matrix elements we derive by perturbation theory the total annihilation

cross-sections

an. rg4 (p (( M) N.R.

oan = 8g2 log 4p l (p?M) E.R.

p being the momentum of the heavy particles. In the laboratory system

where an anti-nucleon with momentum p' andi energy E' collides with a

nucleon at rest the cross-sections are

kan. = 4Mp'' (p' (< 2M) N.R.

20 Equations (3, 15), (3, 18) and (3, 19) of I are incorrect.

21 loc. cit. .



MCCONNEz-Proaduction and Annihilation of Negative Protons-Ill. 65

#an. 4ME' - - 1 (E' ?> 2M) E.R.

This N.R. value of the cross-section agrees with that found by Helstrom.22 The cross-section diverges for small values of p' and decreases as the

energy increases, tending slowly to zero as E' tends to infinity. The mean

life-time in lead of a slow negative proton is now 2-2 x 10-i sec. . This is about one hundred times greater than that obtained in the pseudovector

coupling case, but is 7T7 x 104 times less than the life-time when the annihilation is into photons. So for either coupling the annihilation will

be into pseudoscalar mesons rather than into light quanta.

PART IV.

DISCUSSiON OF A HIGH -ENERGY NUCLEON-NUCLEON COLLISION.

Lord, Fainberg and Schein 23 reported a single nucleon-nucleon colision, the energy of the primary particle being 3 x 1013 e.v. . The number of observed

secondary particles was 15, which is in agreement with Fermi's theory of high

energy nuclear events.2' According to this theory about half the secondaries should be nucleon pairs.

It will be of interest to compare this result with those derived from our

calculations. To obtain the cross-sections for nucleon pair production we

may put E = 3 x 104M in equations (1, 9), (1,14) and (2, 12), and we shall assume that

f2 2 _ he

N 25, A = 04.

Then

a = 2-1 X 104 (-) for pseudovector coupling and perturbation theory,

a= 3 x 10_(t G h

" ,, damping 10-2

a = 6 6 X (I ) ,, pseudoscalar ,, ,, perturbation

The cross-section in the first case is inadmissibly large. When damping effects are included, the probability of pair production in a nucleon-nucleon

collision is quite negligible. According to the pure pseudoscalar theory the

chance of a pair being produced in a nucleon-nucleon colhsion is about 2 per cent.. The result is in every case very different from that which comes

from Fermi's theory.

I am indebted to Professor W. Heitler for helpful comments and to

Mr. A. H. Barrass, B.Sc., for assistance in numerical calculations.

22 loe. dt. . 23 IOC. Cit. . 2* IOC. CU. .

PROC. R.I.A., VOL. 56, SECT. A. [11]



C 0 N T1' FN'I'S

PAGE

SYNGE (J. L.):

1. On the Tranisfer of Energy betweeni Electronmagnetic Dipoles 1

SCHR6DINGER (E.)

2. Electric Charge and Current engenideredi bv comnbined

Maxwell-Einstein-Fields . 13

O'BRIEN (STEPHEN) and SYNGE (J. L.)

3. The instability of the Tippe-Top explained by Sliding

Friction . . . . . . 23

HERIVEL (J. W.) _

4. A General Variational Principle for Dissipative Systems ... 37

MCCONNELL (REV. JAMES)

5. Production anid Annihilation of Negative Protons-Ill ... 45

HERIVEL (J. W.)

6. A General Variational Prinlciple for Dissipative Systems

II. . . . . . . ~~~~~~~67

CORRIGENDA.

Page 56, linie .5.

94Zp fy4p

ForI+ 123ead I

+-12M



production and annihilation of negative protons: iii

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