applicability of the weizsäcker-williams method to meson fields

Applicability of the Weizsäcker-Williams Method to Meson FieldsAuthor(s): James McConnellSource: Proceedings of the Royal Irish Academy. Section A: Mathematical and PhysicalSciences, Vol. 55 (1952/1953), pp. 183-194Published by: Royal Irish AcademyStable URL: http://www.jstor.org/stable/20488555 .

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[ 183 ]

11.

APPLICABILITY OF THE WEIZSACKER-WILLIAMS METHOD

TO MESON FIELDS.

BY REV. JAMES MoCONNELL,

St. Patrick's College, Maynooth.

[Read 26 JANUARY, 1953. Published 15 JULY, 1953.]

CONTENTS.

PAGE

Introductson and summary ... .. .. . 183 1. Meson spectrum of a movmng nucleon . . . 184 2. Dtfferential cross-sections for meson-nucleon scattering ... ... . 18 3. Charged theory of meson produotton by proton-neutron collzstons ... .. 189

4 Charged theory of meson produetzon by proton-proton collisions 192

5. Symmetrcal theory of meson production by nucleon-nucleon collssions ... 193

Introduction and 8ummary.

THE method proposed by C. F. von Weizsicker ' and the late E. J. Williams 2

to resolve the electromagnetic field of a fast electron into a spectrum of photons was first extended by Heitler and Peng 3 to the resolution of the field of a fast nucleon into a spectrum of mesons. This extension has been

very useful for the solution of problems involving nucleon-nucleon collisions

where usually no direct method of ca]culation was available. However, as the method was designed to deal with the electromagnetic field of an

electron, it would be desirable to have some means of checking the reliability of applying it to the meson field of the nucleon.

The direct calculation of the cross-section for production of pseudoscalar r-mesons by a nucleon-nucleon collision given by Morette " provides such

a check. In this calculation perturbation methods were employed and it

was assumed that the mesons have pseudoscalar coupling with the nucleon. It was found that for large values of the energy E df the incoming nucleon

the cross-section for meson production went down to zero with log F E

i C. F. von Weizs?cker, Zeit. f. Phys., 88, 612 (1934). s E. J. Williams, Kgl. Dan. Vid. Selsk., 13, 4 (1935). 3 W. Heitier and H. W. Peng, Proc. R.I.A., A, 49, 101 (1943). * C. Morette, Phys. Rev., 76, 1432 (1949).

PROC. R.I.A., VOL. 55, SECT. A. ( 23 3



184 Proceedings of the Royal Irish Academy.

We have examined the total cross-section for production of charged

pseudoscalar mesons having pseudosealar coupling with the nucleon using

the Weizsacker-Williams method and perturbation theory. Only values of

E greater than 12 Bev were considered but the calculations may easily be

extended to cases of lower energy. Radiation damping may also be included

for a collision between a proton and a neutron provided that a charged meson

theory is adopted.

The results for very high energies are as follows:

(a) for a charge symmetrical theory the cross-section is proportional

to (log E)2 E

(b) for a charged theory and a proton-proton collision the cross

section is proportional to (log E)2 E

(c) for a charged theory and a proton-neutron collision the cross

section is proportional to log E

E

There is a discrepancy with the results of the direct calculation in cases

(a) and (b), though in all cases the cross-section tends to zero when F

becomes very large.

If we compare the two asymptotic values of the cross-section for meson

production in the case of a proton-neutron collision in the charged theory,

we deduce that the minimum value of the impact parameter for the collision

is approximately equal to the Compton wave-length of the proton. The

study of meson production for an energy 3 Bev of the incoming nucleon 5

then gives a value of the coupling constant g such that g 4. hc

1.-Meson spectrum of a moving nucleon.

The problem of the production of nuclear force r-mesons by a nucleon

nucleon collision is reduced to that of meson-nucleon scattering by the

Weizsicker-Williams method. Taking units such that h -_ c _ = 1,

yt being the meson mass, we found5 that a fast proton (neutron) with mass

M and energy E produces a spectrum of positively-charged (negatively

charged) virtual mesons having pseudoscalar coupling with the nucleon,

the number of those with energy in the range (E, e + dE) being

Ire

5 J. McConnell, Proc. R.I.A., A, 55, 169 (1953),



MCCONNELL- Wedzstcker- Willams Method to Meson Plelds. 183

The function D can be approximated by

A_E for_ < 1 E E

D o (2)

- 'cm + C for 'M > I E; E

where A, B, C are constants which depend on the minimum value of the impact parameter for the nucleon-nucleon collision as follows:

bmin. A B C

h ft 1-75 0 2 2 Mc

h 1-2 0 1-2 Mc

___ _____ _ ___ _ ~~~~~~~~~(3) 2 - 0-65 0-15 0-8

3 ft 0-4 0-11 0-48 Mc

4 Mc_ 0-24 0-08 0-3

In a charge symmetrical theory the moving nucleon will be accompanied also by a field of neutral mesons whose spectrum is the same as that just given except that in (1) the coupling constant g is replaced by go, the coupling constant for neutrettos.

2.-Differential cross-sections for meson-nucleon scattering.

The differential cross-section for the scattering of a meson with energy e by a nucleon at rest cannot be given as a single expression valid for all

values of c. We, therefore, distinguish between the non-relativistic (N.R.) and the extreme relativistic (E.R.) cases according as the energy of the meson




is small or large compared with the rest energy of the nucleon. We employ perturbation theory throughout the calculations. In the N.R. case the

differential cross-section for the scattering of a charged meson by a nucleon

is 6

4u112 dcx, (4)

where d t2 is the element of solid angle. The scattering of a neutretto into

a charged meson does not occur in the g4-appoximation. To obtain the differential cross-section for scattering in the E.R. case

we work in the centre-of-mass system where the momenta of the meson and

of the nucleon are equal and opposite. We recall that the matrix elements

for the emission or absorption of a charged meson i li,h energy E0 are, re

spectively,

+ - /7

(iUB P2 UA) (5)

where P2 is the usual Dirac matrix and UA, UB are the spinprs for the nucleon in the initial and final states, respectively. The matrix element for the

emission or absorption of a neutretto is

+ 7- 90 (UB P2 IA),

the plus or minus sign occurring according as the nucleon is a proton or a neutron, respectively.

The scattering of a positive meson by a neutron and of a negative meson by a proton take place through the transitions

Y1 + N - P - Y'+ + N',

Y- + P -)N y- + p,.

It may be deduced from (5) that the compound matrix element for either of these transitions is

Tg%2

-2 (SB* USA

E1

6 Cf. equations (7) and (8) of reference 5.



MCCONnELL- Wezsdeker- Williams Method to Meson Fields. 187

where C is the energy of the incoming meson. The differential cross-section

for the scatteriing is then

94 dc ---32E (1 + cos 6) ds2 (6)

W being the angle between the momentum vectors of the incident and scattered

meson and da the element of solid angle. The scattering by a proton of a positive meson, or of a negative meson

by a neutron occurs as follows:

Y__ + p _)yt + Y'+ + N ---9 Y + + PI,

or

Y- + N- y- + Yt- + p -Yl+N1

The matrix element in either case is

4vg2

M2 + 2j52 + 2 (ii p (UB* HA))

where p and p' are the momentum vectors of the incoming and scattered mesons, respectively. This leads to the cross-section

d q 2M2 + p2 (1 + COS 0) 2 {M2 + 2p2 (l + cos 0)}2

Finally the scattering of a neutretto by a proton into a charged meson occurs through the transitions

Y p + Pt_ P' >y+ + N,

yo + p YO + N' + Y+ -- Y+ + N.

The matrix element is

M2 - 232 + 2(jp)

j2{M2 + 2p2 + 2 (p p')j ingg (uB* UA),

and the scattering cross-section is

a22 (232 ? p2 (1 + COSb)}{M2 - 2j2 (1 - COS Q)}2 d dcr

=

.4

- 32

? 2---(1

? COS

- d -




The cross-section for the scattering of a neutretto by a neutron into a charged

meson has the same value.

For future reference we note how the E.R. cross-sections are transformed

to the laboratory system where the nucleon is at rest. In this system let

p and E be the momentum and energy of the incident meson, p' and c' the

momentum and energy of the scattered meson. We perform a Lorentz

transformation on the assumption that in the centre-of-mass system

p >> , so that we may approximate p and p' by c. If E is the energy of the

moving nucleon, we have

t M ( + p C os )

E

so that approximately

E'= (VM2 + E2 +4 ECos)

E2 + V\/M2 ?+ e2

It follows that

_ 2,E' HE' M 1+ Cos = 2 + ME N

anid, therefore, the range of values 0 to iT of 0 corresponds to

E = e ME' - M- 2E'

Moreover,

d-2 _ 2r(os)d'(02 M dg -2== 2,7d (cos 0) 2= 2X + 2s dE' (1O)



MCCONNELL- Weizsadcker- Williams Method to Meson Fields. 189

3.-Charqed theory of meson production by proton-neutron collisions.

We wish to calculate the total cross-section for meson production when

a proton (neutron) with energy E > 2M2 collides with a neutron (proton)

at rest. Since M - 6-6Qz, the lower limit of the energy is about 12 Bev. We

may regard either nucleon as being at rest and consider the scattering by it

of the virtual mesons of the other. The mesons will appear to be produced

by the collision. The energies of the virtual mesons will range from 1, the

rest energy of the meson, to a value which is less than E and which we take

equal to 2 E. As we are considering only the total cross-section, we take

one of the nucleons at rest and multiply by 2 the resulting cross-section for

scattering. In the charged theory we deduce from (4) a differential cross-section

for the production of a meson with energy in the range (E', E' + dE') by the

mesons of the Weizsacker spectrum whose energies lie between c and e + de

g6 D D #NR dEdE' 2M e E2 dEde'.

The range of variation of e as a function of E' is again

ME'

E M-2E' (11)

Similarly we may deduce from (6) a cross-section on the supposition that E >) M without any restriction being imposed on E'. We may re-write (9)

and (10) as

2 E 47rde'

M E

and equation (6) readily gives the differential cross-section

OdEd I g6D D 4ME 5 {(M + 2E) e - ME} dE dE'.

As E =- M corresponds to e - 2M, we agree to employ ON . or 4 according

as e is less or greater than 2M.

The upper limit 2 E of the energy E of the virtual mesons puts a

restriction on the range (11) of E. Since M 2 attains the value E for M - 2E

E ME 2M + E we have that




E E ......mtf orEr'< ME

31 -2,'2 (M +E)

ME ME

r .. 2 E for c' > 2(M + E)

On plotting the domain of integration in the (c', c) - plane we see that we

ME may integrate first with respect to EC from M + 2- to E and then with

respect to E from 1 to 1 E. Thus the total cross-section is 2

r2M E J E e A

(PP-N- 2{ Dde N.R. dE' + 2J D de{ bdE'. (12) 1 Me 2M Me

M + 2e J+2W

'EM

Since E > 2M2, we have E < 1 for the range (1, 2M) of e in the first E

integral. Hence by (2)

r 2M e gf6A [2M fe dE {SD dE J #N.R. d?E J d _

1Me 2E ~ Me M + 2e M + 2e

g6A = MEz! (2M- 1). (13)

ME In the second integral of (12) we may replace M-+-2c by 2 M and obtain

approximately

E re A g6 d I E tE e

2DM J - 4M 0 - J{(M + 2E) c' - Me} de' J2M Me 4M 2M E lM M + 2e

u6 jAE 1 M>

4M J2M E2 2E3)

B

g6 [M M 1 B

LA + - de

g6 riE I / Mx /1 4M{E{BE(-+ EE 2) +oE2 + ) 2 tI

TM



MCCONNELL- Weizsdcker- Williams Method to Meson Fields. 191

96A E M I 1 Mi

4-4E log i2T19 + 2~ 2~M E )

q6Bf J31 (31 2) - -4-E l og }M 2E

y 6C iM -2 M (MS2 - 4):

4M ; E

?M iE2

The total cross-section is, therefore,

g6A E M2

PP -N - 2E ilog 2M2 + 7d6 - 3

9fiB Mjl 1 l (M - ) 96 + M- 2E2 M 4M log__ 26%M- +

When E )> 2M2, the cross-section approaches the asymptotic value

q6A log E

2 E * (14)

The inclusion of radiation damping does not affect this result very much.

logE A similar -

E - dependence of the cross-section was obtained when the

nesons were assumed to have pseudovector coupling with the nucleon.7

Our result is in qualitative agreement with that of Morette 8 calculated

directly for nucleon-nucleon collisions by the Feynman method When f2

g2 replaces f

in her calculations, we obtain for the cross-section the 4w

asymptotic value

Sg6 log E

MI E * (15)

Equating (14) and (15) and putting M = 66 M ' we deduce that

A = 2*4. (16)

7 J. McConnell, Proc. R.I.A., A, 55, 101 (1953). ? C. Morette, Phys. Rev., 76, 1432 (1949).

PROC. R.IA., VOL. 55, SEUCT. A. [ 24 2




h hi According to (3) bmin = X M gives A = 1-75 and bin = M gives

A = 1-2. Since our calculations are not very accurate, all that we can

deduce from (16) is that bmm11 is of the same order of magnitude as the

Compton wave-length of the proton. A study of the production of pseudoscalar

mesons by a nucleon-nucleon collision for an energy of 3 Bev and a com

parison with experimental cross-sections led to possible combinations of

values of b inj and the coupling constant.9 Applying these results to our

present investigations we may suggest the combination

h 2

bmn.L Xc ' P c 4*

4L-Charged theory of meson production by proton-proton collisions.

The cross-section for production of 7r-mesons by proton-proton or neutron

neutron collisions when E is greater than 2M2 is evaluated as in the previous

section. There is no change in the value of the first integral of (12), and

equation (7) yields

A 34g6D 3M + 2E' D i dE de' =

(H2 ?

4E,E)2 dE dE'.

We then obtain approximately

D Dde J de' 2M Me

M1 + 2e

eg CED e 3M +2,E'de

M 12M J M (M2 + 4E dE

gs LIE; 2,E de = 2M JjR (log - + 3) D

291 2{1 MH1 2 = MJ2MAp (!iog +) dc

Ig6 2 r Q1 1 2,- 3 /l 2e 3 + f[B

- - log ? + - + C 2 log ? 2 dE .

Mi

? J. McConnell, Proc. R.I.A., A, 55, 169 (1953).



MCCONNELL-Weizsdcker- Williams Method to Meson Fields. 193

This and (13) give the total cross-section

- = 9A (log )- + (6 - 2log 1M) log 2M + 7.4 - (log2M)2}

2E { 22/2) ? ( g M)og Mj

6C0 E 2 M- 2' + y log y -j Mlog 1M + (4- log M)

The asymptotic value for E >> 2M2 is

g6A (log E)2 (17)

2V E

This does not agree with Morette's result (15).

5.-Symmetrical theory of meson production by nucleon-nucleon collisions.

A charge symmetrical theory of meson production by a nucleon-nucleon

collision for an energy E > 2M2 requires us to consider the neutrettos in

the spectrum of the moving nucleon. As we are interested in the charged mesons resulting from the collision, we use the scattering cross-section (8)

A to calculate f. The method previously employed gives

D #dco d = _ 904D (3M + 2e') {(M + 2E) E' - 2E2}2 daE'2 JI e4 (M2 + 4EE')2

Since #N.R now vaniishes, the total cross-section for production of charged mesons by the virtual neutrettos is approximately

{iE eA = = 2 J:DdE J de'

2M lM

g2g4 r- 2e 3 o =990 (log - + ) D M 2M M 2 ) 2

that is,



194 Proceedinqs of the Royal Irish Academy.

92 904A ft E\2E _ 9e2E itlog) _- (log 2M)2 + (3 - 2 log IM) log

-

2E 'A MI 2M2

24 D7 (lo _E~2 (lo EV

_ (log _) -(logM) + (3 - 2 loog M) log m

+ a2 0%C 2 o 4

Blog_ + (5 -_2 log M) M 2.

2E ) 3 H 2 H

The asymptotic value of 0. for large E is

g2q04A (log E)2 (18) 2 E

This tends to zero less rapidly than (14), so we may say that (18) is the

asymptotic value of the meson production cross-section for proton-neutron collisions in the charge symmetrical theory. For proton-proton collisions

we add (17) to (18) and obtain the asymptotic cross-section

g2(g4 + g04)A (logE)2 (19) 2 E

Morette's asymptotic value for the cross-section in the symmetrical theory for any nucleon-nucleon collision is

24g6 log E

M E

which differs from both (18) and (19). Thus there is a discrepancy between

Morette's result and ours in the symmetrical nieson theory.

My thanks are due to Professor W. Heitler for valuable criticism.



Page, 26, nineteenth line. rxa

For { read .

Page 171. Equation (1) should read

q (e) de _- dq2 sin2 a Z2 (K12 - K2).

Page 172. Line 4 shotuld read D - Z2 (KI2 - K02) sin2a.

Table I should read

TABLE I-A , B, C as finctions of bmn.

bmrin A B C

h ,1 0 45 0 0 7

Mc 0035 0 045

2 h 0-27 0.05 Y-36

3 M 0418 0.05 0-25

4 h 0-13 0*05 0.19 Me

Page 180. Lines 14, 15, 16 should read "A = 0-45, B = 0, C =0 07

and the total cross-section is 002, which is smaller than the experimental value by a factor 200. The discrepancy ...."

Page 181. The numerical values of 0 should be smaller by approximately a factor 3. The effect of this is to amend Table II as follows

a2

hc 1 4 10

bmin << u- h h 3 h mm 2 ~mce Me Mc

Page 182. It is to be noted that the above corrections do not alter the

result 4.

Page 185. The table is to be corrected as on page 172

Page 192. Lines 1 and 2 should read

"According to (3) bmm . h

gives A O-45 and

bmin. h- gives A - 0-35. Since our



applicability of the weizsäcker-williams method to meson fields

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