Direct pair production by .TI:LECTRONS

DIRECT PAIR PRODUCTION BY .TI:LECTRONS

Theais by

Richard 0 ' Neil Hundley

Jn Partial Fulfillment of the Requirements For the Degree of

Doctor of Philosophy

California Institute of Technology Pasadena, California

1963

ACKNOWLEDGMENTS

The course of this work was supervised by Professor Jon Mathews. whose advice and criticism are gratefully acknowledged.

The numerical ealewations were carried out with the aid of the facilities and staff of the Caltech Computing Center. whose assistance was invaluable.

During the course of this work I was supported by a National Science Foundation Fellowship. and by a Rand Corporation Fellow- ship. I am very grateful for the support of these two fellowships.

ABSTRACT

The total cross section for direct pair production by elec- trons in the field of a heavy nucleus is calculated, using the Born approximation for the nuclear Coulomb field. The calculation im- proves on previous calculations of direct pair production in four

respects: (1) all of the first-order Feynman diagrams for the process are included, (Z) the exchange effect is included, (3) the

effect of screening is treated more accurately using the Thomas· Fermi screening function, (4) the integration of the differential cross section is done with increased accuracy. These improve- ments in the calculation result in a theoretical accuracy of 10-15%

for low Z elements. and of the order of 25% for high Z elements , for which the neglected Coulomb corrections are more important.

The present results are compared with previous theories and with experiments. It is found that there is reasonable agree- ment with experiment for energies up to about 1 Bev, but that for higher energies the experimental situation is too uncertain to allow a definite conclusion.

TABLE OF CONTENTS

PART TITLE PAOE

I L~TRODUCTION 1

II OUTLINE OF CALCULATION 7

IU TREA TWJ!:NT OF SCREENING 15

IV TERMS THAT CANCEL IN TOTAL CROSS SECTION 19 V APPROXI.W..ATION Wl.ADE IN EVALUATING T~~ACES 21 VI .MAGNITUDE OF EXCHANGE TERMS 24 VII RESULTS OF INTEGRATION OVER d3

p+ d3

p_ 28

VIII DISCUSSION OF RESULTS 36

APPENDICES

A. TRACES AND SYMMETRIES 47

B. COVARIANT METHOD OF INTEGRATION 61

C. INTEGRALS 63

D. KlNElVLATlCS AND LIMITS OF INTEGRATION 87 E. DETAILS OF NUMERICAL CALCULATION 95 F . NUW.>.ERICAL RESULTS FOR tr(q} 97

G. NUMERICAL RESULTS FOR cr 107

REFERENCES 113

-1-

I. INTRODUCTION

Direct pair production is the creation of an electron-positron pair by a charged particle as it passes through the electric field of a nucleus. ?ihen the incident particle is an electron, this process ia also called the trident process. The purpose of this thesis is to calculate, using the r.."lethods of quantum electrodynamics, the total cross section for direct pair production by high energy electrons, taking into account effects which have been neglected in previous theoretical discussions of this process.

The original theoretical work on direct pair production was cc::.rried out by a number of authors: notably Niahina, Tomonaga, and Kobayasi (1),

wro

^{used the '·'!eizsacker-Williams m f;thod in which the}

incident charged particle is replaced by an equivalent photon spectrum , which is then combined with the known cross sectior• for ordinary r-~a.ir

production; Racah (2), who used the positron hole th6ory; and Bhabha

f

~.

who considered both the 'deizsacker-'Nilliarns method and the positron hole theory.

Each of these calculations resulted in a series expansion for the total cross section in decreasing powers of ln(E

1/m), where E

1 is the primary energy and m is the electron mass. The leading term, at high

energies, of this expansion was the same in all the calculations, and was, 2 2 . 2 2 2 2 3 /:

for the unscreened case, u

=

z (e /1ic) (e /me ) (28/271T)ln (E

1 m). Although these various calculations obtained the same leading term, they disagreed as to the remaining terms, and this resulted in

-2-

larae m11nerical disagreements. In particular, over a wide range of energies the results of Bhabha and Racah differed by over a factor of three. This disagreement was partially resolved by Block, King and

.vada (4), who showed that it was due to the neglect by Bhabha of certain terms of lower order in ln(E

1Irn} during the integration of the differ- ential cross section. (Specifically, the neglected terms were of order

1 2

( :;;'

I '

^{d 1}

< -· . I

^>

~r.. '-'I rn; an n !!..l m compared to the leading ter:.n of order

l _n 3 < ,-. , ..

I ,

^>

1

m, .

^Block et al. modified the Bhabb.a. cross section to include

th~se terms, and obtained a result which was in much better agreement with t~1at of Racah. Ho·.;,ever, the various theories still disagreed by factors ranging from about 25o/.·, at 100 Bev to about

so·%

at 100 },~:ev.

;t...'i.ore recently, the problem has been disc:.1ssed by Murata, Ueda,

and Tanaka (5), using the methods oi covariant perturbation theory. These authors were primarily interested not in improving the accuracy of the calculation, but rather in understanding, in terms of Feynman diagrams, the approximations made by the previous authors. The m ost significant rcault obtained by Muro~a. !!_ ~· was that all of the previous authors had made approximations which were equivalent to neglecting a certain group of Feynman diagrams. Murata,

!:!

!!!_· estimated that the leading contribution from thc13e omitted diagrams was of the same order as the t<;:,rms omitted by Brobha but included by Ra.cah and Blo~k ~

!!!·

^{As a}

by-product of their work. rv.i:urota et al. redcrived the Bhabha formula fo1· the cross section.

The major defects in these previous theoretical calculationo of the direct pair production cross section may be summarized as follows:

-3-

(1) In the integration of the differential cross section numerous approximations have been made. These approximations,

besides reducing the accuracy of the final result, have resulted in the introduction of various arbitrary parameters, of order of magnitude one, into the final expression for the total cross section. The uncertainty of the values of these para:neters results in an w1certainty in the total cross section of order ln²{E

1/m), compared to the lu::1ding term of order ln³(E

1/m}.

(2) For the case of an electron as the incident particle, all of the previous calculations neglect the exchange effect. In t~rms

of

^{Feynman diag}rams, this is equivalent to neglecting dia- grams a', b', c', and d' shown in Figure 1.

(3) All of the previous calculations also :.~eglect diagrams c and d of Figure 1. Murota !!_ !!!_. (5} estimate that this produces an error in the total cross section of order ln 2 (E

1/m). How- ever, v.te '\vill show later that because o£ cancellations that occur between various terms, the actual error is of order ln (E

1/m).

(4) The screening of the Coulomb field of the nucleus by the!

atomic electrons has been treated only w1der the approxima- tion of "com?lete screening" (3), which is only valid for

E

I z ,

^{1/3 .}

1 me >> 137 /Z where z 1s the nuciear cha1·ge.

:t,·~urota ~ .!!_. (5) concluded on the basis of defects (1) and (3), that for direct pair production by electrons the existing theoretical calculations were unreliable for energies less than 10 Bev.

-4-

p_ p_

(a)

{b)

p_

p_

lc) ^{d)

p_

{ b

I)

p_ p_

X

{ C I)

{d I}

FIG. I FEYNMAN DIAGRAMS FOR DIRECT PAIR PRODUCTION

BY AN ELECTRON.{ NOTATION EXPLAINED IN SECTION li}

-5-

The purpose of the present work is to rec<:.lcula.te the total cross section for direct pair production by electrons, eliminating the defects of the pr€;vious calculations. Specifically, we ·Ni.ll incJ.ude the effects o:J:' the exchange diagra~••s and of cliagrar.r1s c and d, we will improve the

treatr..;~;n·/; of screcnir1g by using an analytic2.l rep1·ceentation of the l"erini- Tho:mas form factor, and we will Lr.prove the accu:racy with 'lnlhich the differential cross section is integrated by resorting to numerical intGgra- tio:'l on a high-speed electronic computor.

In this calculation 'l.ve will, as is custon1ary, :n~glect the nuclear

•

recoil energy , and we ·uill treat the nucleus as thG point source of a Coulo:;:nb field. ·,-..;e will neglect Coulo:mb corrections, and will trea·~ the

Coulo~nb field in first Born approximation ..

**

/.'e will also neglect rudi-

ative cor!'ections. Finally, we will specialize \:o the case where all particle energies are extremely relativistic, and will neglect tern"s of

· <

I_,

^{2 h}

or..:1er m .::.;1 , w_ ere E is the eaergy oi any particle.

In Section II we will present an outline of th~ calculation, and in :3ection III we will discuss our treatment of screening. Certain tern.,s that cancel in the calcl...llation of the total cxoas section are discussed in Section IV, and the app!'oxirtlations made in evaluatkg the traces &N~

discussed in Gection V. The znaguitude of the various exchange tern1.s is discussed in Section VI, and in Section VII the reslllts obtained for the

*

As our results will show, the dominant contribution to the total cross secticm comes frorn the region where the nuclear recoil :r:nomc:mtu::n is of the order of 10-l 1:/Iev/c or less. This corresponds to recoil energies of the order of 1 ev or less.

**

^{In Sectio!' VIII <:te} briefly discuss Coulomb corrections, and estin1ate that they ·.;-Ifill be of the order of 10% for 'he heavy elements.

-6-

cross section after intet;;rating over the mo1~nenta of the created pair are

;:n-esented. In Section VIII the final results of the calculation are p:re- sented and their significance is discussed. Various details of the calcu- lation are presented ir. the ap[,)enclices.

-7-

II. OJTLIN~ OF C.A.LC ULP. Th)N

Treating the Coulomb field in first Born approxim ation, the lowest order Feyn:r:1an diagrama for the direct pair productio? process with a.n electron as the incident particle are shown in Figure 1. In these diagrams we denote the incident electron 4-lno·mentum as pl' the outgoing positron 4-momentunl as p +' and the outgoing electron 4-momenta as p₂ and p _.

Diagrams a', b', c', and d' are the result of the exchange effect.

The m atrix element corresponding to these diagrams will be of

*

the for:m

• a (q)f

111

'11 +

fr ( 1

+

IJ1 fl

+

111

11

11 ^< a b c d

'it ¹'11 ¹'1 IV ¹'11

- (/1/a

+/h b

^{+ " 1 c (2-1)}

(2-2a)

2 2 2

(pl-p2) [ (p+·q)

+

m

1

^(2-2b)

* ····

·•--'e use a metr1c

.

wttn p

. '

•q =

-

p •

-

q - p q , so tnft

-

p₁ 2

=

^p 2 ^=p~_2 ^{= p} 2

=

^-m 2 ^. We are using Wlits such that fl = c :?1°and e /41r-= 1/!37. ' Our Dirac matrices are such that the Dirac equation for electrons is, in momentwr..

9ace, (i-y• p +m)u(p) = 0, and our plane-wave epinor normalization is u(p)u(p)

=

^{1. V is the n}ormalization volume, and q is the 4-momentum contributed by the Coulomb field. (Note that the recoil of the nucleus is -q. }

- \)'-

ii(p _>yl-'-v(p +)u(p

2

h

¹¹[ i"t'· (p

2 -c1} -m] '\'

0 u(p 1) (p++pJ[ (p₂-q) +m ]

( 2-2c}

( 2- 2d)

and the exchc:.nge n-..atrix elements

/h

^{'T) •} etc. are obtair..ed from these by a

the interchange p

2 - p _.

The quantity a"l(q) is given in terms of the external field ATl(x) by

a (q)

=

TJ

f'or a screened Coulo~b field, we will have

A""(x

. ,

} = 3'"

., 411'

₀ - - -Ze

1-; I

^(2-4)

where G(

1-; I)

is the atomic screening fu11c\::ion and Z is the nuclear charge. This gives

a ( ^q)

= c (

^{q '}

o

² e

'f) o' -r;o ( 2^rr)

3

^(2-5)

_,

where the atomic form factor F( q ~) is given by !

1 (2-6)

=

%

\ve will diDcuss our choice for G(

1-; I>

^{in Section lll.}

The matrix element will now be of the forrn

-9-

where

N=

where we denote

/}'J

'•i ""

/J1

'1 +

/J1

^TJ

+ 111

'1 +

111

~,

a b c d

'f.. ^I Ti /ltl I ·n /ltl In /ltl ^1·..., /ltl ... .,

I i I = /II ' I + ^{/1/ .,} + /II 'I +Ill .,

a b c cl

The differential cross section is now

(2-7}

(2-8}

where the summation sign involves an average over initial spins and a awn over final spins,

f3t

is the velocity of the initial electron,· and the density of final states ia

3 3 3 3 V d Pz d p+d p_

pf

=

⁹

(2 _Tr,. \

To facilitate later stages of the calculation, it will be convenient to re- introduce the d4

q integration, so that we have

-10-

We now insert the expressions for

N

^and

Pr

and integrate over all final states. '~Ve muot multiply by a factor of 1/2 for the average over initial spins. Since we will integrate d3

p2 d3

p _ over the er1tire available phase space, we must r.aultiply by another factor of 1/ 2 to correct for the double counting of identical final states. Doi'ng this we obtain for the

*

total cross section

2 ^? 2 6 _d3 3 d3p

Z ^-o.~r ₀ m \ d3q \ p2

~ d P;-) -

('

<r=

2

4~ !3, lr2 ""1r . , ;

^{\ dq}

1T .t..l! ~ ^{v .)} + ' ^{.j 0}

r F(-2)

• I q

i -ol

1.. q ]

2 4

o

(p

1^{+q-p?-p+-p }o(q} )6_ 6.,. •

·~ • 0 'IO n.O

-~

> ( ^{IJ'(fl •;}J}

^{X, •}

^lh

^"1*

^lh

^{I X,)}

• .• ...J

spins

(2-13)

where a = e2

/4Tr

=

^{1/137. 04, and r = e2 /4Trm = 2. 818 x}

w·

^{13cm. ,.,e}

0

note that the term

/J1

'l'l*

/{J

~ ^{in equation 2·13 is} the non-exchange con- tribution, and the term

/J1

',i*

/J1

'X. is the exchange contribution.

At this poini; we are left with the task of evaluaUng the spin sum.- mation in equation 2-13;, and then carrying out the int~gration over the differential croas section. The integration is the most difficult part of the calculation, since it involves seven non-trivial integrations• The procedure ·we follow is to carry out two of the integra.~ions analytically;

using a covariant method of integration, and then do the remaining five integrations nun"lerica.lly. In the remainder of thia section we outline

*

_{In this expression we} have used the fact that !f1'-11 *11tll1 'X.

/J1

11 "" lli, X.

the same amount to the total cross section as ¹ 111 ,

l^{y; ..... f:. 1\, • •• /)If}

"*

^/111 tx_

" ., 111 wlll contru.,utc the same a:.-:nount as 1 ' 111 •

will contribute and that

-11-

the steps in this procedure, the details of which are presented ⁱⁿ the following sections.

and

To simplify the discussion, ^we introduce the notation

•

Ilj>-.

⁶

-~·-~

lh

^{11* lrJ ~}

=

^m

_/...)

_{i J}

spins

I ' r·l >-.

⁶

~--~ ....

r ·~

. . . ,.

ij

=

^m

_L

'" i IIi·

spins J

The total cross section now becomes

v=

~ \~

\ dq 0 ( q } 5 0"' I

I

K

,.~

.. 0 0 r'IO 1\.0 \ :...1 lJ

ij

(2-14)

(2-15)

(2-16}

Tl~ ·,~

The traces, i. e. the I ij and I ij , are evaluated in Appendix A.

In that appendix we also discuss the various symmetries that exist between

,~ ''1~

the 1

1. and I.. • One of the results of these symmetries is that certain

J lJ

•

_{In these} expressions the subscripts ^• ij denote the diagram. The factor

6 ..,~ ¹"ll~ '\'}~ ¹1"1~

m makes the lij , I ij , Kij , and K ij dimensionless.

-12-

terrns in the total cross section cancel out. This cancellation is diG- cus sed in Section IV.

In evaluating the traces we n'lake certain approximations which involve dropping terms of order m 2

j E2

compared to unity, where E ie one of the energies F . ^{These app}roximations are dis- cussed in Section V.

·,.;e inake certain other appro::dr.nations with respect to the e:Kcha.nge

terzns, i. e. the ^{t 'r'l} 1\. . • • .-. •

I . : • These approxtmahons are discussed tn .:.ectton VI.

lJ

The next step in the calculation is to evaluate the

K :'.A.

and

K '.r:~ .

lJ lJ

These integrals are evaluated analytically by a !'ll.ethod which takea ad-

. *

vantage of their covart&"lt form. The integrals are transforr.>1ed to a special Lorentz systen1 in \ovhich they become relatively easy to evaluate, and then the resultant expressions are transformed back into the labora- tory coordinate system. Thia method of evaluating the integrals is il- lustrated in Appendix B and the results obtained for the various integrals a:i"e presented in Appendix C. The final results obtained for the

K~

_lJ

~

and

K'JX.

are

pres~nted

^{in Section VII.}

Having evaluated the c13 p+ d3

p_ integrals, i.e. the

K~.x.

^and lJ

K':·]'A., we make use of the 6 6 r) 6(c )dq and then car"!"'tr out the rc•

lJ '110

X.o .

¹

o o -

³

. . . i 11

**

rnatntng tntegrals nurner ca y. To do this, we write

*It i8 for this reason that the

Ilj~

^{and I}

'i ] X.

we1·e kept in

a

covariant fo!""t".n..

**

^';Vc

n o·~e

^{that the d3p_ d3 p _} integrals are the easiest to evaluate ana-

lytically. Due to 'i:he structure of the denominators in the matrix clement2, 3 3

a.Jr!y othe:.- COlnblnation, such as d p.>d p , would be r~"luch more difficult

.!... - -

~o evaluate analytically.

-13-

(Z-17)

.1\ 1\

.... ... - -

^{1\ 1\ cos}

^a

2 and The limits on

these integrals are determined from the ldt1.e1&-1a.tics of the process. This is discus sed in Appendix D.

To facilitate the nwnerical worl:, we write the cross section as

u =

where

u(q}

7 2 2 2

""' a. r • 0

"\'

i >

K'!.o

\ w

^lJ

ij

12

J

^{4Y(q) (2-W)}

(2-19)

(/1/e leave the integral over q Wltil last, beca\lse this is the only integral

tlmt involves the screening form factor, and therefore the only one that is a function of Z. By leaving the q btegral until last, we can obtain

t~'le screened cross section for different values of Z "vi.thout repeating all of the previous integra.ti01~s.

In .l~ppendix E we discuss certain details of the numerical rnethods used, and we give an estimate oi the numerical errors involved. The re- sults obtained for o-(q} are tabulated in Appendix F :for various values

.r. .,.,.. d

*

o:.: .. :..

1 an q. In thai Appendix we also tai:mlate.. various combinations The tabulation of cr(q) may be useful at a later date for calculating the screened cross section at values of Z not included here, or with a different screenin;; function.

- - - -

-14-

of terms, such as non-exchange and exchange, that enter i:1to cr(<i).

The final results obtained for cr, as a function of E

1 and Z, are tabu- lated in l\ppendix G.

In Section VIII "ve summarize the results of the calculation, and compare these results •vith previous theories ane "vith experiments. v'/e also diseuse the overall accuracy of the present calculation.

-15-

III. TREA Tl-1lENT OF SCREENING

F'or an accurate calculation of the direct pair production eros::~

section, one must inclucie the screm'ling effect of the atomic GlectronG on the nuclear Coulor ... '"lb :field. To gain a qualitative idee:. of the impor- tance o£ sc::-eenh1g, we note ti:mt, as shown in .P:..ppendix D, the min.irr.u:;.'l:

nuclear recoil 1no;·;nentun1 is 2 q . :: <.im / El.

:nu::. 'l'his corresponds to a

the aton"lic electron cloud i3, using the Thoma.s-l"l'n·•'>"!i ;:nodel, of the ., . d .,/ "L,l/ 3

orai\:Jr or = ... r.c.ie :£.. • \'"/e wo•lld now expect that SC1'0ening will. have a significani: effect for _r > -_-

d.

max ^{'This corresponds to energies such} .. .., > 1/3

u~.at .:::.1 - 13 7m/Z ' • This qualitative argument showe that sc:rzening shouh! have an ir.nportant eitcct for the energieo, 100 }/lev to 10 Bev, that we are prh-narily interested in.

To i::1clucle th•:.: effect of screening in an accurate manner, one :o:nus'i: :!'eplacc the ol·d.im:~ol'Y Coulornb Held by a !:lcreened Coulomb field, c. f. ccuation 2-4, -,;:;he:re the at:o:mic scrGening iu...Ylction is

G(r~ 4w r-r 2

·- 1 -

:z- \

^{p(:r'}r' d:r'}

_) ~j

(3-0

·;;rh.3 re p(r '} is the electron density distribution. ;f'/e are now faced ·with

th~ problem of choosin&, a m odel for the electron density distribution of

The ::nost accurate atomic models available are the Hartree- E'ock m od'71• which inclades correlation and exchange effects, and the Hartree :rr.odel, which includes correlation effects, but no exchange effects. Fo:r the ptu•poses o£ this calculation, bo·I.YJeve:r, both of these

-16-

znodcls suffer from two disadvantat;es: results for the ator:::~ic sc!"eening function are available only for a few values of Z, and these results are

available only in tabular form , which n·.~ake therr" avvi-':ward to use in a numerical calculation on a computer. Because of these disadvantages,

we will use instead in this calculation the Thomaa-Fer1.·.ni atomic :moclel.

This n"lodel neglects both correlation and exchange effects, but gives a reasonably accurate representation of the atomic elect:ron cloud, omitting, however, the details of the shell str ucture. For our present purposee, the Thon-:<as•Fern-:.i m.odel :bas the advantage that the ato:rnic screer.ing iu..r'lction is known for ail values of Z, and that aeveral highly accurate

.I

analytical representations of Tho::nas-Fcrn1i results are available. In this calculation we will use the !•£oliere representation of the Tholr.tas- Ferm i screening function.

G(r)

wh.e :re

a=

'\'

3

=

_LJ ^I

i=l

121 a e

i

n"l Z

i/3

and the coefficients are

al

=

^0.1

bl = 6

b.r

1

a

az =

b 2

-

o .

55 1. 2

*

This is (6}

a .... _;;

=

^{G. 35}

b3 ^::

o .

3

(3-2)

(3-3)

(3 -4)

*

^{'Ne note} that Royental (7) has developed a very simila1· representation of the Thom.as-Fermi screening fu.1ction. The difference between the

~v!·.oliere and I!oyental representations would ?roduce a difference in our

final result of the o:rder of 0. 1%.

Usbg this expression !or the result

\ ' 3

=

L

i=l

-17-

C(r), we have for F(q ), 2

a i

2

b. \ 2

q + l ..l.

\ a I

cf. equation 2-6,

(3- 5)

F rorn the form of equation 3-2 we see that the most important contribu- tions to the cross section will come from the region r / a - 1. By com.- paring the Moliere representation with the exact value of the Thoma:3- Fermi screening function as tabulated by I'\.obayashi, ~

!!.·

(8), we find 'i:ha·, the error in the Moliere representation is lees than 0. 2'7~ in the region 0

s

r/a :S 6.

The most important error arising iron"l our choice of a screening function will be that due to the inherent inaccuracy of the Thornas-Fermi n-:odel. To obtain ^n&t estimate of this error, we have used the graphical cl.ata of Nelms and Oppenheim (9) to compare the Thomas -Fermi form factor to the Iiartree form factor. For the value oi recoil momentum q = :mZ l/3 /13 7, which is the region of maximum contribution to the total cross section, we found th;.;: results shown in the following table:

z

6 (C) 26 (Fe) 80 (Hg~

Ratio of Form Factors Thomas -Fermi/Hartree

0. 95 0. 97 1. 05

These results indicate that, r~lative to the Hartree form factor, the Thomas-Fermi form factor introduces an error of the order of 5%.

Since the cross section involves the square of the form factor, this will

-lB-

introduce a lO~'v error in the final result for the total cross section.

It should also be noted that the Hartree and Hartree-Fock form factors are expected to differ by a few percent (9/.

-19-

IV. TERMS THAT CANCEL IN TOT...-'\.L CROSS SECTION

In Appendix A certain symmetry relations a1·e obtained connecting .,). • X.

various of the I i. and 1 .~ • Among these is the relation that m.'lder the

J lJ

intercl'..a.nge p+ ~ p_,

(4-1)

and

-

^(4-2}

Thic symmetry bal:J a particular significance for the calculation of the total cross section. To see this, we note that K:t.). is defined by

lJ

3 3

r .

^{d P+}

r

^{d P_}

K~.X.

= ,

'J ,}

.t<~

+

j E -

4 ~x.

6 (p +q-p -? . -p )1.1.

1 2 - ·r - lJ ^(4-3)

From thi3 definition ar:.d th;~ eyrro..n1ctry relations 4-1 and 4-2 it follows iJ:nmecliately that

K '!'lX.

ac

= - ^~~

^(4-4)

and

K·tlX.

.. ad

=

^{• K'1X.} "bd ^{(4- S)}

Thia :result has the consequence that the non-e;,.;change interference te:rzo,."lS between diagrams a or b and c or Cl cancel each other out and clo not contribute to the final result for the total cross section.

Besides simplifying the calculation, this cancellation is of interest in conjunction with catim..ates made by rv~urota, ~~· (5), of the erro1• in- volved in completely neglecting diagrams c and d. They showed that

-20-

the leading term in the contribution to the total cross section from the square of diagram s a and b was, a1n~.rt frorn constant factors, of order

7

ln.:;I(E

1/ n"1). They also showed that the leading term coming from the :.:~quare of diagram s c and d was of orde:r ln(£

1/m). They then inferred irom these results that the leading contribution coming from the non-ex- change interference term s between diagrams a or b and c or d should be of the order ln2

(F

1/ m).

The results we obtained in equations L1-4 and 4-5 show that, while these interference terms may rnal;:.e a contribution to the differential cross section of thia relative order of magnitude, they in fact do n_ot con- tribute to the total eros s section. This means that, as long as one does not consider the exchange effect, the error involved in con~pletely neg- lecting diagrams c and d is of order ln(E

1/m), and not of order

ln

²

( ~: 1 ^{/rn }}

^{as suggested by Murota,}

^~

^!!_.

^*

*

^{The p}resent calculation will still contain correction tel'ms to the pre- vious calculations of order ln2(E

1/m). These will come from the in- creased accuracy of integrai:ion and elimination of arbitrary param eters in the final result.

-21-

V. APPROXIMATION !viA DE IN EVAL u;. TING TRACES

The calculation of the traces can be appreciably simplified be- cause of the limitation to extremely relativistic energies. This simpli- f'ication is possible because of the fact that at v~ry high energies the dominant contribution to the total cross section comes from small

!

angles about the forward direction. Thie "small-angle effect" is a iam iliar feature of all electromagnetic processes in the ultra-relativis- tic energy region. and for direct pair production it was pointed out in the original theoretical work (3.10). Before discuasing the approxima- tion that we make based on the small-angle effect. we will digress briefly to give a qualitative explanation of its occur rence.

The concentration of the total cross section at very small for- ward angles is due to the form of the various denominators in the matrix elen:.enta. i.e. the Feynrnan propagators. These propagators have poles at angles slightly beyond the forward direction (cos 9 > 1). and in the lim it of infinite energy these poles approach the physical region.

To see explicitly what we are talking about. consider one of the denominator factors. for example (p_ ..

+

p )2

• Vie have

'

-

For ::n/ E + << 1. rn/E _ << 1. this becom es l(-:-' ' "'""' ,2

2 m .t·-+ ...- ^.~:.._,

(p++ P) =-- - - -

-;-" ~

• 2E , Z (1 • COS S ^{.L )}

T - r-

-~+--

(5-1)

(5-2}

-22-

We see fron1 this expression that the pole occurs for

cos 3 -:·- ^(5-3)

We also see from equation 5-2 that a terrn containing this deuom.inator factor will be large only in the 1·egion 6+- $ (m2

/E+EJ¹

f

^{2, or what is}

the same thing, in the region

Similar considerations hold for the various other denominator factors. Vle see from this qualitative discussion that the cross section

·;;trill be large only for srr;.all angles in the forward direction, as has been

i_:>oint~d out previously (3, 5,10}. >ie also see that this has the conse-

quence that the dominant contribution occurs in the region where all of

~he

^{four-vector dot products are of the order of m 2, i.e.} P( p

2- m 2 ,

- 2 t

?+ m , e c.

He.ving discussed thi8 s:mall-angle effect, let us now consider the fo:rzl'"l of tl}e various terms arising in the evaluation o1.the spin summation, i. e. the ~races. As can be seen from the expressions in Appendix A, a typical term is of the forrn

whe:re Pa, pb' etc. are various four-mor.nenta, the f-1• v indices are to be summed over, and the )'"11, "{). arise from the interaction with the external field. ';'/hen this expression is evaluated, and the 'If, ). indices are set equal to zero, corresponding to a static external field, cf. equa- tion 2-16, the result will contain terms of the two types:

-23- Type 1

Type 2

The terms of type 1 ariae from "V "1 and "V).. contracting on two of the four-momentmn vectors in the trace, and the terr.-"lS of type 2 arise from

':} )..

'l and "V contracting on each other.

'Ne now see that. because of the concentration of the cross section at small angles. terms of type 2 will be of order m2

/ E2

com - pared to term s o£ type 1. In the evaluation of the traces in Appendix P:, we will neglect the terms of type 2. This will appreciably simplify the calculation of the traces.

-24-

VI. ~.f:AGNITUDE OF EXCHANGE TI:Rlv~:'3

1\'iurota,

!:! !:.!.·

^{(5), discussed the magnitude of the} exchange terrns, and concluded that they were of order (m/E

1)2 ln2

(E1/m) relative to the leading; non-exchange terms and therefore negligible for m/ E

1 << l.

However, they only considered the e:cchan;:;e terms arising frorn diagrams

2. an~ b in Figure 1. In actual fact, the exchange te rm.s arising from diagra1ns c and d, or from d'le interferende te:rn1s between diagrar.ns a or b and c or d, are Zi""lilCh larger than the above estimate, and con- t:.·ibute an appreciable amoant to the final result.

In O!'der to unde1·stanu qualital:ively the difference in magnitude of ·~h.~ various exchaa;c terms, it is useful to view the exchange effect

a~ a :repulsion in the phase space of the t\uo final state electrons. Cme can say that, if in the absenc<O: oi e;:(change the two electrons tend to con'le off tosether in pha3e space, then the exchange effect vli.ll be large. How- ever, if in the absence of exchange the two electrons tend to come off far apart in phase space, then the exchange effect will be Sl'!'lall.

In our present problem the bulk of the total cross section col-neG fl·orn small forward angles, cf. section V. Therefore we can suppose that ~he elect!"ons come o:If at the same angle, and the magnitude of the exchange effect will the~'l depend only on their energies. ·:He can then aay:

(1) If in the absence of exchange the two electron energies tend to be equal, then the exchange effeC'i: will be large;

(2) If in the absence of exchange the two electron energies tend to be very different, then the exchange effect will be small.

·//e can now use this approach to w1.derstand the difference in magnitude

-25-

of the various exchange terma in the present problcn-n.

Consider first diagram b. Th~ two denominator factors of this diagram are

and

In the forwarci direction we find, using E

1 >> rn, etc. ,

2( . .,. .,_._ . 2

., r.n -'~1

-

^r:. 2 i

< r>t·

P2r ·· = ^(o-1)

and

2("''' ... ' ( \2 , 2 :na .t:.l- ~-2¹ p -1 p -2 ') " -. "\"'t''---\ = - - - - --~ ·-- .. -

.:.:.11:.?..;.;.-

(6-2/

de see that the product of these two factore will be a r.li:i.'limum., and the contribution to the cross section a rrlaximum, when !!::

2- L

1 and

r~ << E

2: Thus, with respect to diagram b, th•J energie:l of the two final :itate electrono tend to be very different, and the exc!~·.ange effect will be emall. A similar result can be obtained for diag;c.·am a.

Now let us con~icler diagram c. The two denominator factors arc

(p++ P) 2 ^and

In ths forward direction we find, using E

1 >> m, e:tc.

(r .~ .. _.. ·. 2 =

.1:.1+' i:' -· ., ... ^·-~"

"

..

~-· +--

(6-3)

=

-26-

m 2

[ E ( -· + r:: )2 +...,. (E + ^"'G' ,2

2 l!.. + ~ - .1::. - 2 ^J..:..

+

¹

(6-4)

'.'/e see that the product of these two factors will be minimum, and the contributior1 to the crosa eection J;naxin'lwn, when E+ ""'

E _-

^:2₂ ^-

E / 3 ·

Thus, wi.th respect to diagram c, the two final state electrons tend to have the same energy. and the exchange effect 'Nill therefore be large.

Consideration of diagram d leads to a. similar result.

The above argLunents enable us to understand qualitatively the difference in magnitud-e of the various exchange terms.

,v·e

can under- stand why. as point-ed out by 1\·~urota, ~

!!.!.·.

the e:::cha.nge terms arisi••g from diagrams a and b are very small. Cin the othel" hand. the above results indicate that th:e exchange terms arising from cliagrams c or d should be much larger.

The arguments uaed above do not give a quantitative estimate of the magnitude of the vaTious exchange tern1.s. However they suggest that for thooe cases where the final state electrons tend to have the oan'l.e energy, the exchange ter1ns should have roughly the same magnitude as the non- exchange terms. With respect to the terms aTising from diagrams c and d, this hnJ.'>lies that the ~xchange terms should be of order ln(E1/r.a), as are the non-exchange terms (5). ·:lith respect "o the interX'erence terms between diagrams a or b and c or d, the situation is nluch less clear, botb because ~hese terms contain denominator factors which are a mixture of the two types discussed above, and because the non-

-27-

exchange ter:ns, due to certain symmetries, cancel each other out.

~However a plausible estimate of the magnitude of the exchange inter- ference terms is that they will be roughly of the order of magnitude that the non-exchange interference terms would have been if they had not cancelled, namely ln 2 ^(E₁/m}.

The above discussion indicates that the total exchange contribu- tion to the cross section will be rnuch larger than the estimate of Murota,

~ !!.·•

^{which was (m/ E}₁^)2tn2(E₁^{/m}, and which waa obtained on the baaia}

of considering only diagrams a and b. The discussion indicates that the exchange contribution will in fact be dor..'1inated by the exchange terms involvin;; diagrams c and d and by the exchange interference tezoma between diagrams a or b and c or d. These dominant e~change

terms are estimated to be of order ln(E 2

1/m) or ln (E

1/m).

The qualitative arguments of this section are confirmed by the detailed numerical results of the present calculation. From the resulta presented in. Appendices F and G and au1-nmarized in Section VIII, we can see that the leading exchange terms are much larger than the Murota estimate, and are in fact of order ln(E/ m).

We close this section by noting that in the ,emaL"'lder of the calcu- lation we will neglect the exchange terms arising solely from diagra:ms

d b . th t K¹TI~ K'i1X. v''ll~ K¹"1~ ^. 1 b a an , 1. e. e erma aa , bb , r.. ab , ba' atnce, as a 10wn y :Murota, ~

.!'!!.•

^{and by} the arguments in this section, they are certainly negligible, being only of order (m/E

1}2ln2(E

1/m).

-28-

In this section we present: the results obtained for the KijX. and

1t'j).. 3 3

K after integrating over d p .1-d p_. This calculation was carried

-· ij .

' • n).. d 1¹'11).. •

out stG< :rting wtth the expressions obtatned for the I:. an i. 1n

lJ J

Appendix A, and using the results for the various in~egrals given in Appendix C.

In the r esults presented in this section we have, after the inte- gration, set '1

=

^~

=

^{0 and q} _o

=

^{0, since these are the only terms con-}

tributing to the final result. (See equation .Z-16.} 'v'o/e use the notation

I

The coefficients A., B., D., D.,

1 1 1 1

F., Gi' V ., ·~v. which appear in the reaul~s arise aa a result ox the co-

t 1 1

variant integration pxocesa, and are presented in App2ndix C.

We present the results for the non-e,~change terma first, and then the reaults for the e;~change terma.

1^{.~00 -}

·" a a

-

A. Non-Exchange Terms

m 2 11' 2

6 ^A """2 { B - 2 B .t. rn ( B - B ·'· 2 .B \

\ .,j:!..l' 11 17'

T

3 1 · 15'

-29-

Koo _ m 2 11'

ab - Z( ..~..

2)2

P(Pz · m

4wm2

~..-00 - E 1 J A { 'ti' [ k2( • k • 2) 2 2 • k

"'·cc -

4 20 2 \

^{.r-~o ~·1} P2 -r m - m Pz

k (q

+

2pl• q)

+

E

2m²(c/+ 2p 1• q)}

and

-30-

·;.:-00 - 2lm'l 2

l .,. 2 ( "' ^{It )}

'-·cc: -

₁ ^{4 ' ...:.k .nl - ... ,.2}

K

K~~ ^{is obtained from} K~~ by the interchange p1 - Pz• q - -q •

.. , f ^1' 1 · d. " · A d. A K⁰⁰ K>.oo

.';.:;ccause o t,'le symme~!"Y re attons tscussea tn ppen tx , bb

=

'"'-aa •

The re:maining non-exchange terms (K⁰⁰ , etc.} cancel out, as is dis- ac

cussed in Section IV.

B. Exchange Terms

I

+ (V 4p

2·k - 2V

5

rr/+ zv

₆^)[E₂^(q2•2p₁^•q)-

2E

₁^{(p( p}₂^{+ p}₂^•q)])

-32-

' 21Tln 2

K oo =

--oro__.---""""""---

dc • 2( 2+., )( 2 ₂ ,

t< q '"·Pl• q q - Pz•qJ

• \_ V

0 {m²[ Ef (q• k-p

2• q)

+F~ ( pl •

^{q+q• k) -E}₁^Ek(p₁^{• q+p}₂^{• q) +E} ₂^{Ekq• k}}

c 2 2 2 . 1 2 2]'

+E1E

2Lm (q•l;•k -2p!"p

2-2m -2p(k-2p

2· k}+z k q j +Vl {ElEk[ 2p2• q(pl• k+q•lc.+:m2)+2m2P].. q-q2pz• k]

+::-~[

^{q2(Pl· P2 +m 2> ·Zpl· q Pz· q]}

+2E 1E

2( (p

1• k+q•

k)(p 2 ^{o! ~ -q · k+ 2r~"l}

²

⁾

^{+p1• k Pz • k] -2m 2q•}

^{k ( E:+E~)}

+E2Et-J 2p

1• q(q• k-p2• k)-q2

p1• k-2m2

q. k} }

+V 2 {ZE1E2[(pl• Pz -{-pz• q)(pz• k-q• k}+pl• Pz Pz• k

2 2 1

+m (2pl•p2+pl•q+pz•q-2m _-r~q·k)]

+E2Ek[ q2(pl• Pz.+mz)+pz.• q(m2·2pl• q)]

2 2 2

+ElEk[ Pz• q(2pl" p 2+2pz• q-m )+q m ]

2 2 2 2 . 2 2 }

+E2[2p

1•q(q·k-p

2·k)-q p

1· k-m (2p

1•q+2q +q· &:)]-2E

1m p 2·q

2 2

+V 3 {2E1E

2(k q• k-Zp 1• kp

2• k)+E

1Ek[ q p2• k·2p

2• q(pl" k+q• k)]

+E2Ek( q2pt• k+2p( q(p

2• k-q•lt/]

+E~[

^{2pl•qp2· q-q2(Pl'Pz+n/)]}}

2 2

+V 4 {ElE2[ q• k(pl• P2 +m +p2• q)+pz• k(T ·Pz" q- 2Pt" Pz>+²P( km 2 ]

2 2

+E

2

^{E~( P( q{p2• q-m }- -\-<P( Pz +2m )]} 2

2 2 2

+

E~

^[

~

^{Pfk+pl•q{pz·k-q·k)]- I.'lE'k[q2m} +p2•q(pl"p2+p2•q)]}

+ v 5E2m2 [ El(4pl• Pz +2pz• q-q2)-Ez(2plo q+q2)) + 2V

6[ E 1E

2(2q2

+p₁• q-p 2• q-2p

1•

p 2 ^{)+E ~}

^p1•

^{q- E~}

^{Pz" q])}

-33-

'

K oo _ dd -

z z

2 2

+ m (Zp 1• p

2- p

1• q-k -6m -i·4p

2• k+4cr k-2q )]

z z z

+m [Et<Pz•q+q·k-q

>-

^{Ek(pl•q+p2· q+ T )] }}

and

-34-

2

+ 2B1Ek( E2^(p1• q-q•

~-2p 1 ^•

^{k+Zp2• k+k2}

^-zrn

²⁾⁺

^~

^{E1 +p2• qEk]}

2 2

+W

0[ 2E

2^Ek(2p1•q-p

1· k- 3

z )

^-2E₁^F₂^{(;.. +r:112)-m 2Ek(Ek +2E}₁^)]

-35-

'oo 'oo 'oo

Thee terms Kda , Kca , and K cb are obtained from the above expressions by the substitutions

V - V!i . i

Under this substitution ^we have

K 'oo

cc ^{- K}

'

^oo

da

'

- K oo

ca 'oo _ K'oo

Kdd cb

The remaining exchange terms (K a a. 'oo , etc.) are neglected, eince, as indicated in Section VI, they are expected to be negligible.

-36-

Vlll. DISCUSSION OF RESULTS

The analytical expressions obtained in the previous sections were integrated numerically over the remaining part of the final state phase space to obtain the final result for the direct pair production cross section. The numerical calculation was carried out for various values of Z and for nine values of primary energy between 10 Mev and 100 Bev. The results of this calculation, both for the total cross section and for various terms that contribute to it, are tabulated in Appendix G. In this section we will summarize these results and discuss their significance.

In Table 1 we present numerical results for both the un- screened and screened total cross sections. The table shows the behavior of the cross section as a function of primary energy, and also shows the effect of screening. As has been shown by previous calculations (1, 3, 5), we see that the screening effect begins to be noticeable at about 100 l~ev, and becomes of increasing importance as the primary energy is increased above that value.

In Table 2 we present results for the exchange contribution to the total cross section, crEx

1 , and for the non-exchange con-

. c 1ange

tribution from diagrams c and d, a CD' We see tl-:tat u CD becomes negligible, i.e.' less than one per cent of the total contribution, for energies greater than or of the order of 100 Mev, and that crExchange becomes negligible for energies greater tl'lan or of the order of 1 Dev.

The results in Table 2 are of interest because of the dis- agreement with the estimates of Murota, et al. (5), These authors

Prit:<1ary Energy

(Bev)

0.01 0.03

o.

1 0.3 1.0

3.0 10.0 30.0

!00.0

-37-

Table 1

Total Cross Section for Direct Pair Production

Total Cross Section (in units of

z

^{2 c:2 r} ₀²⁾

Unscreened

z

^{= 26}

1. 070 1.068

5.798 5.753

19.89 19.51

43.73 41. 95

79.87 72. l'l:

121.

s

100.9

194.6 118.5

273.2 133.6

384.2 156.6

( 2 2 . . . -30 2

t! r ₀

=

^{4. 22S x 10 em )}

:3creened

z =

⁸²

1.066 5.713

19.24

41.02

69.37 94.90 106.7 116.2 131.2

-38- Table 2

Effect of Exchange and of Diagrams c and d

Primary Unscreened Screened

Energy

z

^{= 82}

(Bev)

, I

u

c rJ

^{cr Total 0" Exchange C1 Total cr}

cr/

^{u Total ff-~xc ~ h ange}

^I

^{r:r T o t} al

0.01 0.0506 0.1620 0.0505 0. 1600

0. 1 0.0074 0.0173 0.0072 0.0148

l.O 0.0023 0.0069 0.0020 0.0047

10.0 0.0009 0.0053 0. \1006 0.0031

100.0 0.0003

--

^0.0001

--

c:rCD = Non-exchange contribution from diagrams c and d cr Exchange = Total exchange contribution (absolute magnitude)

-39-

estimated that 0' CD would be much larger than we have found it to be.

and that O'E xc ange h · would be much smaller. As we have discussed in Sections IV and VI, this disagreement arises because of the cancella- tion of the non-exchange interference terrns between diagrams a, b and c, d, which Murota, !!_ al. were not aware of, and because these previous authors considered only those exchange diagrams arising

solely from diagrams a and b.

Table 2 also shows that the dominant contribution arises from the non-exchange terms involving diagrams a and b, and that for energies of the order of or greater than 100 Mev the contribution from all other terms is less than ten per cent of the total.

In Figures 2 and 3 we compare our results with those of pre- . al 1 . •

vtoua c cu atlons . This comparison is of particular interest be- cause .of the disagreement among these earlier ca~.culations. As we indicated in Section I, this disagreement ia primarily due to the var- ious approximations made by each author during the integration of the differential cross section, and also due to the various arbitrary

parameters that these approximations introduced. The present work may be thought of as an attempt, by means of an accurate numerical calculation, to discriminate between the results of the previous cal- culations.

The unscreened erose section is presented in Figure Z. For energies up to 1 Bev the results of th' present calculation agree

*

^{The sm}all circles in Figures 2 and 3 are approximately of the size of the numerical errors in the present calculation.

1000.0.---~----~--------------------~~~--~----~~~~ FIG. 2

DIRECT PAIR PRODUCTION CROSS SECTION

1----+----1

UNSCREENED ----+--l ~

(1"

7 1 /Ffi IOO . O~== =t ====t==t=t~ ==~~~~ ~~fj~::~~~~~~~ -- _J ____ j__j_j _

2 2 2 Z a ro

· -

10.0

AW

1/ I

y A/ _LL/'

Raca~V 1.0 0.01

I ;: lock ~~ hino ..

0.1 1.0 Energy In Bev

0 This Calculation 2 2 -30 2 ar0 =4.228xl0 em 10.0 100.0

I ~ 0 I

1000.0 100.0 (S" Z2 2 2 a r0 10.0

/ v

I.O 0.01

/ v

1:.1

FIG.3 DIRECT PAIR PRODUCTION CROSS SECTION

/ v

SCREENED

z

= 82 Bhabha-Muroto\

v- ....--

__... ... __... ... ~

v v '/

"' /

~Nishina / I I I

I 0.1 1.0 Energy In Bev

--- ~ ~ ----

^1-

---- ---

f...--"'

~ v- v v

0 0 /

v /

<Zl This Calculation 2 2 -30 ar0-4.228xl0 em 2 --·- 10.0

-- ----

1-- 1-- ( I ~ ... I 100.0