252
Le!ters to the Editor
Radiative Corrections to Anomalous
Magnetic Moment of Nucleon in
Pseudoscalar Meson Theory
K. Nakabayasi and I. Sato
PhY8ical Indif!uie, Fac'ulty oj Science,
T6hoku Univer8ity
February 12, 1951
The anomalous magnetic moment of
nucleon has been hitherto calculated in the
lowest order by many authors,!) choosing in
particular pseudoscalar meson which seems
to be most promising from some evidences.2)
The result was that the moment ratio of
neutron to proton moments was too large in
absolute value compared with the experiment.
In view of this discrepancy, we thus estimated
the first-order radiative corrections using
Feynman-Dyson method, in case of pseudoscalar coupling. (The coupling constants are
denoted as {1,={1~={1 (charged meson), {13
(neutral meson to be added in symmetrical
theory) and {14 (purely neutral meson) ; they
are equal to the conventional ones J0 2/47rfic
in Heaviside units divided by 27r). The
diagrams contributing to the moment in the
fourth order are shown in Fig. 1. II and
III are reducible to the second-order ones,
IIo and IIIo, responsible for the moment in
the lowest order due to necleon and meson
currents, respectively. After renormalization,
we obtain as the radiative corrections in
nuclear magneton units
JJl=0.472·! [({13 + (14)2'Z"p+4{1({14 - (13)'Z""],
-0.242·
t [{ ({13+ (14)2+ 2{1({14 - (13) }'Z"p+ 2{1({1~ - (13)'Z""]'
/.llla=
Letters to the Editor
.
:
A
. . ,.<...
,
.
253
.
.
1'"
"
----
----
.
-,
I
'
,_ ..."
III f,
\\
,;..
tJ \\
\
I
III c
;.
I
\
«
)
)
I
I
,'~•
,
)
I
,
.
,
I
I
\,
.
I~' ,
,
t
----
\
tee,
:~,
III ..
,, ,
I
(
A
--;..,
,,i\ \
.
I~: Ila~: ~,,:Il b
«
\
\
\
' .... "
--"
I/Jd
I
J
;..
I
«
L,
,
,
,
,
Fig. 1. Feynman-Dyson diagrams contributing to first-order radiative corrections for
anomalous magnetic moment of nucleon. The full, broken and dotted lines denote
nucleon, meson and photon lines, respectively.
Jll1b= -0.077·
H28+8~+84) [(8a + 84) rp + 28r,,].
Jlllc= -0.128· [(8a+8.\)2rp +28 2 r,,],
JlJld=0.626·
! [{48 2 + (8a+84J2}rp+48(8a+84)r,,],
Jllle= -0.084·t8(28-8a-84) (rp-r,,),
Jlll1a=0.089. t8(8a - 84) (r p- r,,),
JllJlb=0.213·!8(28+8a+84) (rp-r,,),
JlIJJc=0.079. 8 2 (r,,-rn),
JllJld= -0.1578 2 (rp-r,,)
taking 7!-meson mass as 275me, while the
lowest-order moments are
JlJlo= -0.237[(8~+84)rp+28rn],
Jlmo=0.348(rp- r,,).
The anomalous moments up to the fourth
order for various theories are tabulated in
Table 1. The radiative corrections are clearly
in direction to remove the discrepancy existing in the lowest order, and this tendency
accentuates itself muchly as neutral meson
is mixed. These features peculiar to pseudoscalar model are interpretable as follows:
Since nucleon-meson coupling takes place
through odd operator rs, the necleon acquires
large recoil whenever emits a virtual meson.
This meson, too, gains large momentum and
energy ;;CMc2 , M being nucleon. mass. This
implies that the distribution of meson cloud
around the nucleon is highly asymmetric or
is associated with angular momentum of
order n. Angular momentum conservation
now requires that the nucleon ought to
undergo spin umklapping. This is the reason
why nucleon contributions in the lowest order
and from l Ia , b, c are not small and negative,
where the nucleon interacts with the external
electromagnetic field after emitting only one
meson. In 1 and lId, the nucleon emits
however two mesons and recovers its spin
direction, thus behaving normally so as to
remedy the above unfavorable effect. And
this radiative corrections turn out to be most
predominant. On the other hand, the meson
current contributions remain to be relatively
small, owing to relativistic increase of meson
inertia.
The values of coupling constant to reprorluce the experimental moment-ratio and
the resulting neutron moments are also
included in Table 1. The purely charged
theory is to be rejected, as requires negatively
large 8. Thus the role of neutral mesons
254
L:?tters to the Editor
tJ·,
tJp
Theory
2nd order
4th order
2nd order
4th order
Charged
0
+0.3488
+ 0.271~2
- 0.02482
-0.4748
-0.3488
- 0.29282
+ 0.024~2
Symmetrical
-0.2378
+0.34813
+ 0.3038 2
+ 0.025132
-0.47413
-0.3488
+ 0.1548 2
-0.4748
+0.3488
+ 0.21082
+ 0.02982
-0.4748
-0.3488
+ 0.21582
Symm. + neutral
- 0.02582
-0.02982
~
--100
4.09
(tJ ,= -6.35)
2.14
(tJ,,= -0.91)
Table I. Anomalous moments calculated up to the fourth order, in nuclear magneton units,
and values of the coupling constants to give e"perimental moment-ratio -1.07. For tJp and
tJn, the first (second) line corresponds to nucleon (meson) current contribution.
is decisively important; it is even possible
to give the correct moments' both for neutron
and proton, adjusting 8.l with 83=8, that is,
taking 8=2.49 and 8~=1.37.
This co:.rpling constants correspond to
the conventional one of order 10, being
quite reasonable from other data.
Thus the radiative corrections are quite
large, so higher order ones must be also
taken into account, which shows however
invalidity of the weak coupling approximation.
Full accounts will soon appear in Science
Reports, Tahoku Uriiverslty, first series, vol.
34, no. 4.
1)
2)
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Phys. Rev. 76 (1949), 1; K. Sawada, Prog.
Thllor. PhlS. 4 (1949), 532.
Z. Koba, T. Kotani and S. Nakai, Prog.
Theor. Phys. 5 (1950), 137, 494; Soryushiron Kenklu 2 (1950), no. 5, 63 & no. 6, 91;
G. Araki, Prog. Theor. Phys. 5 (1950), 507 ;
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