548
Letters to the Editor
On the Soft-Photon Radiative
Corrections to High-Energy
Electron-Electron Sea ttering
Kazuo
BABA
Department of Physics
Nara Women's Universiy, Nara
December 21, 1964
In 1960 the cross sections for high-energy
electron-electron scattering including radiative corrections were calculated by Tsai 0
under the intent to test whether the electron
has any finite size or if quantum electrodynamics is valid at small distances by
comparing the results of the current theory
with experimental data. His work seems
to he the most complete and reliable one
among ·similar calculations for the electronelectron scattering which had been carried
out up to that time. We have, however,
found some small terms to be added to
his results concerning the soft-photon part
of inelastic radiative corrections. These
additional terms depend only on the scattering angle 0, ,but not on the incident energy
E so that in spite of their smallness we
cannot neglect them at all for the reason
of the high-energy approximation.
In this short note we shall report on
these additional terms. Almost all notations or abbreviations shall be the same
as in the Tsai's paper which will be hereafter referred to as T.
The contribution to the differential cross
p'
2
k
and similar
diagrams.
Fig. 1. The diagrams for inelastic radiative
corrections.
549
Letters to the Editor
section for the scattering in C. M. frame
dt1 1 from the soft-photon emission diagrams
(Fig. 1.) can be written down as the 1st
expression of Eq. (22) in T:
2
2 !L)
with .:1(0)=~~
12 + ln(sin !L)ln(cos
2
2
u
0)
~ - 1-- ( sin 2"-+cos 2" +2:E
•-1 (2n) 2
2
2
where each l; stands for the contribution
±rom the i-th term respectively in the
bracket of Eq. (1). Now according to our
integral calculations on both photon angle
and its energy variable, each l; becomes
as follows:
•
This corresponds to the 2nd expression of
Eq. (22) in T, and the term .:1(0) is just
the additi~nal term above mentioned. The
semi-qualitative behavior of the function
J(U) is shown in Fig. 2.
)
J_ln 2 2
2
90°-e 180°
0°
Fig. 2. The semi-qualitative behavior of J(O).
+__!__ ln2 (~~)-~}-~]
2
m
2
6
12
Thus we obtain the sum da~lastic+da!ort
corresponding to Eq. (23) in T:
'
dtl~lsot!c+da!ott=d!Jl'
E ) ln( -----;;;:t2
-q 2 )
Ia=--4a
ir:- [ ln( ifE
X
---z
1 { ( -qz ) ( mz )
ln -----;;;:t2 ln ~
dm2 (
-----sE2
s4+q'4 +-s-4_)
q4
q2qt2
11 ( -q2)
X [ 1- 4a{23
-- -+.:1(0)--ln
rc
18.
12
m2
+ln(- .d~ )(ln( ;::;::
)-r)}]
(2)
+terms obtained by interchange
+2 ~
.":::1
1
(2n)
2
. 29a u
sm 2
q2
J'
l 4 =terms obtained by interchange sinf
~cos ~
and substitution
q 2 ~q' 2
in / 3 •
Substituting these expressions into Eq. (1),
we obtain the following result:
~
q'2.
As illustrated in Fig. 2, the numerical value
of .:1(0) is quite small in the neighbourhood
of 0=0° and 180°, but it amounts to
(l/2)ln2 2~0.24=4/18 at 0=:90° which can
not be neglected compared with the numerical term 23/18 in Eq. (2).
Thus, corresponding to Eq. (24) in T,
550
Letters to the Editor
we have
4a {--+J(O)--ln
23
11 ( -q
X [ 1----)
7r
18
12
m2
2
=da~<f>ller [1 +D!.,rtCU)],
with
11 (.-q
-+ t.~ (0)--ln
12
m-
1
4a { 23
u..so·'
r ({})= - - 1r
18
+ln(
k
)(1n(
2
4
)
2
~:!:
)-1)}.
Finally we have
=da~<Piler [1 +D!on(O)+O'~ard ({})]
=da~<f>ller [1 +0'1 (0)].
Therefore the contribution from J({}) to
0'1 (0) is - (4a/7r)J(O). Thus, for instance,
we obtain -(4a/7r)J.(90°)::::::-0.22X10- 2 •
Hence instead of the numerical value in T
corresponding to E=500 MeV
0 1(90°) = ( -13.8+4.3) X lQ- 2 = -9.5X lQ-2 ,
we obtain a slightly larger value
0'1 (90°) = (-13.8+4.3-0.2)
X
10- 2 = -9.7 X 10- 2 •
In view of the physical importance of
the experiment these results should be
taken into account in the analysis of the
colliding beam experiment which shall be
performed in future.
Although the usual type of experiment in
which the target electron is initially at rest
has already been performed by Dally 2l in
1961, the radiative corrections are comparatively small from the experimental conditions, thus our J({}) term has not any
physical importance in this case.
1) Y. S. Tsai, Phys. Rev. 120 (1960), 269.
2) E. B. Dally, Phys. Rev. 123 (1961), 1840.