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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]
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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).
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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*]
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48 Proceedings of the Royal Irish Academy.
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
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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.
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50 Proceedings of the Royal Irish Academy.
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.
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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.
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52 Proceedings of the Royal Irish Academy.
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
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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.
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54 Proceedings of the Royal Irish Academy.
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).
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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
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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
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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
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58 Proceedings of the Royal Irish Academy.
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..
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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
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60 Proceedings of the Royal Irish Academy.
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
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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]
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62 Proceedings of the Royal Irish Academy.
(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.
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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).
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64 Proceedings of the Royal Irish Academy.
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. .
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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]
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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
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