1941] _On Yukawa's Theory of the Beta-Disintegration. 291
(the first term in (28) being ignored), which settles an upper limit for the proper lifetime of the meson. This in good agreement with the value 2•~10-6sec. obtained from the cosmic ray observations, if we choose
correspoding to a cutting off Wave number of the order of 3ƒÈ I think, this result must be considered as very satisfaetory, since through
outthe quantum theory reasonable results can only be obtained by the perturbation method if we cut off at wave length of this order of
In conelusion, I wish to express my cordinal thanks to Professor H Yukawa for his kind interest in this work. Institute of Theoretical Physics, Kyoto Imperial University.
(Reecived Feb. 12, 1941.)
On Yukaua's Theory of the Beta-Disintegration and the Lifetime of the Meson.
By Shoichi SAKATA.
(Read Jan 18, 1941.)
Introduction.
Ina recent paper by M. Taketani and the present author(1) a matrix formulation of the vector meson theory was described. Though this formalism is inadequate for the diseussion of the general character (e.g., Lorentz invarianey, ete.) of the theory, it proves to be very useful for the practical calenlation. Especially, the spin sunmmation of the meson can then be carried out automatically by using the orthogonality
, the theory was developed for the case when the electromagnetic interaction alone is involved. In the first part of the present paper. an extension to the generalnc case when the mesons interact. with the.
(1) S. Sakata and M. Taketani, Scient.Pap. I.P.C.R.38. No. 984 (1940),1, referred as A. See also M. Taketani and S. Sakata, Proc. Phys.-Math. Soc.Japan. 22 (1940), 757
Inmagnitude. A 292 Shoichi SAKATA. [Vol. 23
heavy and light particles is given(2). It is further shown that the present formalism can be easily extended so as to include all the cases of the mesons with spin 0 or 1 considered by Kemmer(3).
In the subsequem parts of this paper, a new derivation of thebeta -disintegration formula and the meson lifetime, resulting from each
of the four types of the meson theories, is given by using the above formalism. The well-known discrepancy(4) between the calculated and
experimental values of the meson lifetime is also discussed.
I. General Theory.
1. Vector Theory. . In the vector theory of the meson developed
by Yukawa and others(5) the total Hamiltonian for the system consist ingof nusons, heavy particles and light particles was given byH=
Hu+Hh+Hl+H'q+H'q+H'q2+H'qq'+H'q'2, (1) with
Hu=•ç1/4ƒÎ(FF)+1/ƒÈ2divFdivF+(UU)+1/ƒÈ2(curlU)(curlU)/dv .
( 2)Hh=•çƒµ{-ihcƒÏ(h)1ƒÐ<•¨>hgrad+grad+ƒÏ(h)3Mre2}ƒµdv+•çƒµ{-ihcƒÏ1hƒÐ<•¨>(h)grad+ƒÏ(h)3MNc2}ƒµdr, (3)+•çƒÓ{-ihcƒÏ(h)1ƒÐ<•¨>(l)grad+ƒÏl3ƒÊ_??_2}ƒÓdv, (4)H'_??_=•ç{-g1/ƒÈ1M_??_divF-g1/ƒÈ(MU)+g2/ƒÈ2(SeurlU)-g2/ƒÈ(TF)}dv
+comp. conj., (5)
H'0'=•ç{-g'1/ƒÈ2M'0divF-g'1/ƒÈ(M'U)+g'2/ƒÈ2(S'eurlU)-g'2/ƒÈ(T'F)}dv,+comp. conj., (6)
H'_??_2=4ƒÎ/ƒÈ2•ç{g2_??_M_??_M_??_+g2/2(SS)}dv.(7)
2) An equivalent formulation based on Kemmer's theory (Proc. Roy Soc. A 173(1939
), 91)has been given by A.H. Wilson (Proc. Camb, Phil. Soc. XXXVI (1940), 363),b
ut this is not so convenient as ours owiny to the singular character of the ƒÀ-matrices.
(3) N. Kemmer. Proc Roy. Soc. A 166 (1938), 127.
(4) L. Nordheim, Phys. Rev. 55 (1939), 506.(5) H
. Yokawa, S. Sakata, M. Kobayasi and M. Taketani, Proc. Phys.-Math. Soc.
Japan, 20 (1938), 720 which will be referred as IV . In the following, we shall use thesame notations as in IV. 1941] On Yukawa's Theory of the Beta-Disintegration293H'g_??_=4ƒÎ/ƒÈ2•ç{g1g'1(M0M'0+M'0M0)+g2g'2((SS')+(S'S))}dv, (8)H'g'2=4ƒÎ/ƒÈ2•ç{g'12M'0M'0+g'2S'S')}dv, (9)
Amongthese expressions. (2), (3) and (4) are the energies of the
mes ons, heavy particles and light particles in the absemce of external
fields respectively. while the rest represents the interaetion energies
betweenthese partieles. U<•¨>, U<•¨>. F<•¨> and F<•¨> are the field variables of the
meson field satisfying commutation relations[Fi(r<•¨>,t).Uk(r'<•¨>,t)]=i4ƒÎhckƒÂ(r<•¨>-r'<•¨>)ƒÂik,[Fi(r<•¨>,t).Uk(r'<•¨>,t)]=i4ƒÎhckƒÂ(r<•¨>-r'<•¨>)ƒÂik,} (10)*
a nd ƒµ. ƒ³, ƒÕ and ƒÓ denote Dirae's functions for the proton,
(M<•¨>.M0) and (M'<•¨>,M'0), and the six vectors, (S,<•¨>T<•¨>) and (S'<•¨>,T'<•¨>), are defined by the following equations:M<•¨>=ƒ³ƒÏ(h)1ƒÐ<•¨>(h)ƒµ, M0-ƒ³ƒµ,S<•¨>=ƒ³ƒÏ(h)_??_ƒÐ<•¨>(h)ƒµ, T<•¨>=-ƒ³ƒÏ(h)2ƒÐ<•¨>(h)ƒµ,} (11)M'<•¨>=ƒµƒÏ(l)1ƒÐ<•¨>(l)ƒÓ. M'0=ƒµƒÓ.S'<•¨>=ƒµƒÏ(l)3ƒÐ<•¨>(l)ƒÓ. T<•¨>=-ƒµƒÏ(l)2ƒÐ<•¨>(l)ƒÓ.} (12)**
where1, ƒÏ2,ƒÏ ƒÏ3. and ƒÐ<•¨> are the usual Dirae operators and the superscript
(h) or (l) refers to the heavy particle or the light particle. The constants
(g1. g2) and (g'1. g'2), which have each the dimension of an electric
charge, determine the strength of interactions of the mesons with
the heavy and light particles, respectively.F inally, m_??_=hƒÈ_??_e, M_??_, M_??_, m and ƒÊ denote the masses of the
, proton, neutron, electron and neutrino, respectivelu.Recently Taketani
•and the anthor(1) developed a matrix
formulation of the vector meson theory for the case when the mesons
interact only with an electromagnetic field Introducing a six-componentwave functionƒÔ-1/•ã
A more general assmuption involving the derivatives of the wave function
for the light particle, which was made in IV order to obtain energy distribution
of the beta-ray of Konopinski-Uhlenbeck type, leads to a very much too small value
for the meson lifetime and is therefore, not considered here.
(***) In order to avoid confusion, we write ƒÔ instead of ƒµ in A, all other notations being the same as in A
meson(*)neutron. [A, elcetron B]=AB-B].(**) and neutrino, respectively. Further, the four vectors. 294 shoichi SAKATA. [Vol. 23
they rewrote (2) and (10) in the following, forms:
Hu=•çƒÔƒÏ2HƒÔdv,(15)
[{ƒÔ(r<•¨>,t)ƒÏ2{ƒÊ, ƒÔƒË(r'<•¨>,t)]=-ƒÂƒÊƒËƒÐ(r<•¨>r'<•¨>), (16) where the operator H is defined byƒÏ2H=hcƒÈ[1-graddiv/2ƒÈ+ƒÏ3{(ƒÏ<•¨>grad)2/ƒÈ2-graddiv/2ƒÈ2}], (17)
ƒÐ<•¨> =(ƒÐ1, ƒÐ2, ƒÐ3).
Further, they also showed that the wave function ƒÔ satisfies in the absence of the external fields the equation
ih•ÝƒÔ/•Ýt-HƒÔ=0. (18)
(*) No confusion between these matrices and the Dirac operators will arise if it is kept in mind that the latter have superscripts 1941] On Yukawa's Theory of the Beta-Disintegration. 295
This formulation can be easily extended to the general case when the
interactions with the heavy and light particles are involved . To write
the interaction energies (5) and (6) in terms of ƒÔ , we introduce rowmatrices
ƒÌ and ƒÌ', which are defined byƒÌ=(-g1/ƒÈM0grad-g2T, g1M-g2/ƒÈS_??_grad), (19)ƒÌ=(-g'1/ƒÈM'0grad-g'2T', g'1M'-g'2/ƒÈS'_??_grad). (20)
these notations, we obtain from (5) and (6) the expressionH'g=-•ã<4ƒÎhc/ƒÈ>•çƒÌƒÔdv+comp. conj., (21)H'g'=-•ã<4ƒÎhc/ƒÈ>•çƒÌ'ƒÔdv+comp. conj., (22)
Now, we go to the momentum space of the mesons. By using the
plane wave solutions of (18), we expand ƒÔ and ƒÔ as follows ((22) in A)
ƒÔ=‡”
ponents, is the solution of the equation
and where the constant, E and p<•¨> satisfy the relation
For a given momentum p<•¨>/c, there are six independent solutions of (25),
which we have distinguished by ƒÉ and ƒÃ ƒÉ=1, 0, -1 and ƒÃ=E/|E|-1,
or -1 correspond to three polarization states, and two charge states of
the meson, respectively. The u's have to be normalized so that ((18) in
A) u<•¨>
(p<•¨>,ƒÉ,ƒÃ)ƒÏ2u(p<•¨>,ƒÉ',ƒÃ')=ƒÃƒÂAƒÉ'ƒÂƒÃƒÃ', (26)
from which we obtain the converse orthogonality relations
‡”<ƒÉ>‡”<ƒÃ>ƒÃ{u(p<•¨>,ƒÉ,ƒÃ)ƒÏ2}ƒÊuƒË(p<•¨>,ƒÉ,ƒÃ)=ƒÂƒÊƒË,‡”<ƒÉ>‡”<ƒÃ>ƒÃuƒÊ(p<•¨>,ƒÉ,ƒÃ){ƒÏ2ƒÊ(p<•¨>,ƒÉ,ƒÃ)}ƒË=ƒÂƒÊƒË.} (27) The explicit form of u may be written as
Using Shoichi SAKATA. [Vol. 23
)-p<•¨>/p. e<•¨>(p<•¨>, 1) and e<•¨>(p<•¨>. -1) are unit vectors perpendicular
to each other.
The as and a*s are_q-numbers satisfying the commutation rela
tions((23) in A)
[ a(p<•¨>,ƒÉ,ƒÃ), a*(p<•¨>ƒÉ',ƒÃ')]=ƒÃƒÂAƒÉ'ƒÂƒÃƒÃ'. (29)Further, N+(p<•¨>,ƒÉ)=a*(p<•¨>ƒÉ,1) a(p<•¨>ƒÉ,1) or N-(p<•¨>,ƒÉ)=a(p<•¨>,ƒÉ,-1)
a*(p<•¨>ƒÉ,-1) c an be•interpreted as the number of the mesons with the
charge. c or -c in the state of energy |E|, momentum p<•¨>/c or -p<•¨>/c
and polarization ƒÉ. Hence, a*(p<•¨>,ƒÉ,1) and a(p<•¨>,ƒÉ,1) denote the
operators which increase the number N+(p<•¨>,ƒÉ) and N-(p<•¨>,ƒÉ) by one
res pectively, whereas a(p<•¨>,ƒÉ,1) and a*(p<•¨>,ƒÉ,-1) denote those whichdecrease them by one respectively.
Inserting(23) in (15). (21) and. (22), we obtain the expressions
Hu=‡”
H'g=-•ã<4ƒÎhc/ƒÈ>‡”
H'g'=-•ã4ƒÎhc/ƒÈ‡”
ƒÌp<•¨>(r<•¨>)=(-ig1M0p<•¨>/muc2-g2T,g1M-ig2S•~p<•¨>/muc2),(33)ƒÌ'r<•¨>(r<•¨>)=(-ig'1M'0p<•¨>/muc2-g'2T',g'1M'-ig'2S'•~p<•¨>/muc2). (34)
296where c<•¨>(p<•¨>,0 1941] On Yukawa's Theory of the Beta-Disintegration. 297
From (31) or (32), we can immediately obtain the matrix elements for the emission and the absorption of a meson by a heavy or light parti cle. The whole of the theory so far developed applies not only to the vector theory but also to the psevdovector thcory considered by Kemmer(3), the only difference being the explicit forms of M, M0, S, T, and M', M'0, S', T'. In the latter theory, these quantities must be replaced by the following expressions:
M=ƒ³ƒÐ<•¨>(h)ƒµ, M0=ƒ³ƒÏ1(h)ƒµ,S=ƒ³ƒÏ2(h)ƒÐ<•¨>(h)ƒµ, T=ƒ³ƒÏ3(h)ƒÐ<•¨>(h)ƒµ,} (35)andM=ƒµƒÐ<•¨>(l)ƒÓ, M'0=ƒµƒÏ(l)1ƒÓ,S'=ƒµƒÏ(l)2ƒÐ<•¨>(l)ƒÓ, T'=ƒµƒÏ(l)3ƒÐ<•¨>(l)ƒÓ.} (36) 2. Scalar Theory. The matrix formulation made, in the preceding subsection can also be extended to the scalar meson theory studied by
Yukawa and Sakata(6). In this theory, the total Hamiltonian is again given by (1), while (2), (5), (6), (7),and (8) (9) must be replaced by
the following expressions respectively:Hu=•ç{4ƒÎc2U+U+1/4ƒÎ(gradU, gradU)+ƒÈ2/4ƒÎUU}dv, (37)H'g=•ç{f1(WU+UW)+f2/ƒÈ((gradU,M)+(M,gradU)+4ƒÎcM0U++4ƒÎcU+M0)}dv, (38)H'g'=•ç{f'1(W'U+UW')+f'2/ƒÈ((gradU,M')+(M',gradU)+4ƒÎcM'0U++4ƒÎcU+M'0)}dv, (39)H'g2=4ƒÎf_??_2/ƒÈ2•çM0M0dv, (40)H'gg'=4ƒÎf2f'2/ƒÈ2•ç(M0M'0+M0M'0)dv, (41)H'g'2=4ƒÎf'22/ƒÈ2•çM'0M'0dv. (42)
Here U and U are the scalar functions describing the in meson field, from which U+ and U+ are derived by the equations
(6) H. Yukawa and S. Sakata, Proc. Phys.-Math. Soc. Japan, 19 (1937), 1084,referred as II. In II, our f1 was written as g, whereas the interaction involving f2 was omitted 298 Shoichi SAKATA. [Vol. 33U+=1/4ƒÎc2•ÝU/•Ýt,U+=1/4ƒÎc2•ÝU/•Ýt.} (43)
The commutation relations between these variables are given by
[U(r<•¨>, t), U+(r<•¨>, t)]=ih ƒÂ(r<•¨>-t'<•¨>,)
[U(r<•¨>, t),U+(r'<•¨>, t)]=ih ƒÂ(r<•¨>-t'<•¨>).} (44) The scalar quantities W and W' denoteW=ƒ³ƒÏ(h)3ƒµ, (45)
. (46)
The constants (f1, f2) and (f'1, f'2), having each the dimension of an
electric charge, determine the strength of the interaction of the scalar
mesons with the heavy and light particles, respectively. All other
notations have the same meaning as in the preceding subsection.
Now, if we define a two component wave function ƒÔ by
ƒÔ=•ã<ƒÈ/4ƒÎhc>(U/-4ƒÎcU+), (47) operators ƒÏ1, ƒÏ2, ƒÏ3 acting on this ƒÔ by
ƒÏ1=(0 1 1 0), ƒÏ2=(0 -i i 0), ƒÏ3=(1 0 0 -1), (48) and row-matrices ƒÌ and ƒÌ' byƒÌ=(-f1W-f2(M,grad), -f2M0), (49)ƒÌ'=(-f'1W'-f'2(M',grad), -f'2M'0), (50)
( 37), (38), (39), and (44) become again the forms:Hu=•çƒÔƒÏ2HƒÔdv, (51)H'g=-•ã<4ƒÎhc/ƒÈ>•çƒÌƒÔdv+comp. conj., (52)
H'g'=-•ã<4ƒÎhc/ƒÈ>•çƒÌ'ƒÔdv+comp.(53) conj.,
[{ƒÔ(r<•¨>,t)ƒÏ2}ƒÊ, ƒÔƒË(r'<•¨>,t)]=-ƒÂƒÊƒËƒÂ(r<•¨>-r'<•¨>),(54) with
ƒÏ2H=hcƒÈ{1-grad div/2ƒÈ2-ƒÏ3grad div/2ƒÈ2}. (55) The transformation to the momentum space can also be performed
W'=ƒµƒÏ(l)3ƒÓ 1941] On Yukawa's Theory of the Beta-Disintegration . 299
in the similar manner. We expand ƒÔ and ƒÔ in the plane waves asƒÔ=‡”
where p and ƒÃ have the same meaning as before, while u (p, ƒÃ) is now the solution of the equationEƒÏ2u={p2/2muc2+muc2+ƒÏ3p2/2muc2}u, (57)
and satisfies the following orthogonality relations:u(p<•¨>,ƒÃ)ƒÏ2u(p<•¨>,ƒÃ')=ƒÃƒÂƒÃƒÃ',‡”<ƒÃ>ƒÃ{u(p<•¨>,ƒÃ)ƒÏ2}ƒÊuƒË(p<•¨>,ƒÃ)=ƒÂƒÊƒË,‡”<ƒÃ>ƒÃu_??_(p<•¨>,ƒÃ){ƒÏ2u(p<•¨>,ƒÃ)}ƒË=ƒÂƒÊƒË.} (58)
The explicit form of u may be written asu1=iƒÃ•ã
The a's and a*'s satisfying the commutation relation
[a(p<•¨>,ƒÃ), a*(p<•¨>, ƒÃ')]=ƒÃƒÂƒÃƒÃ', (60)
denote the operators similar to a(p<•¨>, ƒÉ, ƒÃ) and a*(p<•¨>, ƒÉ, ƒÃ) in the pre
cedingsubsection.
Inserting (56) in (51), (52) and (53), we obtain
Hu=‡”
As before, (62) and (63) correspond to the emission or the absorption
of a scalar meson by a heavy and light particle, respectively.
Furthermore, replacing W. W', M, M0, M', M'0, by
W=ƒ³ƒÏ(h)2ƒµ,(67)W'=ƒµƒÏ(l)2ƒÓ, (68)
, M0, M', M0 respectively, the above formula can also be applied to
the pscudoscalar meson theory.
II. The Theory of Beta-Decay.
As was first pointed out by Fermi(7), the process of beta-disintegration
can be considered as a transformation of a neutron in the nucle us into a proton with the sintultaneous transition of the light
particle from a neutrino state of negative energy to an electron state of positive energy. In Yukawa's theory, this process takes place in
two different ways: (1) by direct interaction H'gg' between the heavy
and light particles and (2) by the indirect interaction through the
meson field which is pictured as
Neutron•¨Proton+Negative Meson,
Ne gative Meson+Neutrino•¨Electron,
or (69)
Neutrino•¨Electron+Positive Meson,
Positive Meson+Neutron•¨Proton.
The matrix element HƒÀ for the beta-decay process is therefore the sum
of those for the direct and indirect interactions which we shall denote
as H(1)ƒÀ and H(2)ƒÀ, respectively.
The calculation of H(1)ƒÀ is very simple and gives immediately the
espressions:H(1)ƒÀ=4ƒÎ/ƒÈ2•ç{g1g'1M0M'0+g2g'2(S,S')}dv, (Vector Theory) (70a)H(1)ƒÀ=4ƒÎ/ƒÈ2•ç{g1g'1M0M'0+g2g'2(T,T')}dv,(Pseudovector Theory) (70b)*H(1)ƒÀ=4ƒÎf2f'2/ƒÈ2•çM0M'0dv, (Scalar Theory)(70c)(7) E. Fermi, ZS. f. Phys. 88 (1934), 161.
(*) In this case, the invariant quantity-4ƒÎ/ƒÈ2•ç{(g2S+g2'S')(g2'S+g2'S')-(g2T+g2'T')(g2T+g2'T')}dv must be added to the total Hamiltonian, in order to avoid the occurrence of the
static interaction of the ƒÂ-function type.
M 1934] On Yukawa's Theory of the Beta-Disintegration . 301
H(1)ƒÀ=4ƒÎf2f'2/ƒÈ2•çM0M'0dv. (Psendoscalar Theory)On (70b) the other hand, H(2)ƒÀ is given by
H(2)ƒÀ=‡”{(F|H'_??_'|I)(I|H'_??_|A)/EA-EI+(F|H'_??_|I)(I|H'_??_'|A)/EA-EH}, (71) where A and F denote the initial and final states respectively and I
and H the two intermediate states corresponding to the two schemes
of the interactions as shown in (69). EA, EI and EH are the energies
of the states A, I and H, respectively. The summation must be taken
over all values of the momentum p<•¨> and the Mates of polarization, ƒÉ, of
the meson which is emitted in the intermediate state. The matrix
el ements (F, |H'_??_'|I). etc. can be easily obtained from the equations (31)
and (32) or (62) and (63), and are given by(I|H'g|A)=-•ã<4ƒÎhc/ƒÈ>•çƒÌp<•¨>(_??_<•¨>h)u(p<•¨>,ƒÉ,-1)eip<•¨>r<•¨>h/hcdvh,(F|H'g|H)=-•ã<4ƒÎhc/ƒÈ>•çƒÌp<•¨>(_??_<•¨>h)u(p<•¨>,ƒÉ,-1)eip<•¨>r<•¨>h/hcdvh,(F|H'g|I)=-•ã<4ƒÎhc/ƒÈ>•çu(p<•¨>,ƒÉ,-1)ƒÌ'p<•¨>(_??_<•¨>_??_)e-ip<•¨>r<•¨>_??_/hcdv_??_,(H|H'g'|A)=-•ã<4ƒÎhc/ƒÈ>•çu(p<•¨>,ƒÉ,1)ƒÌ'p<•¨>(_??_<•¨>_??_)e-ip<•¨>r<•¨>_??_/hcdv_??_.} (72)
(In the cases of the scalar and pseudoscalar theory, u(p<•¨>,ƒÉ,•}1) must
be replaced by u(p<•¨>,•}1)). Inserting (72) in (71), we obtainH(2)ƒÀ=4ƒÎhc/ƒÈ•èdvhdvl‡”<_??_<•¨>,ƒÉ>eip<•¨>r<•¨>h-r<•¨>l_??_/hc(u(p<•¨>,ƒÉ,-1)ƒÌ'_??_<•¨>(r<•¨>_??_))(ƒÌ_??_<•¨>(r<•¨>_??_)u(p<•¨>,ƒÉ,-1))/W0-_??_p2+m2uc4
(ƒÌ_??_<•¨>(r<•¨>_??_)u(p<•¨>,ƒÉ,1))(u(p<•¨>,ƒÉ,1)ƒÌ'_??_<•¨>(r<•¨>_??_))_??_/W0+_??_p2+m2uc4 (73) where W0 is the energy difference between the states of the neutron and proton. Since W0 is always small compared with muc2 in the
ordinary beta-decay , H(2)ƒÀ becomes approximately
H(2)ƒÀ=-4ƒÎhc/ƒÈ•èdrhdri‡”
summations in (75) can be easily performed with, the help of the con
verseorthogonality relation, (27) or (58). The results are given by 302 Shoichi SAKATA. [Vol. 23
X=muc2/•ã
By using the explicit forms of ƒÌp<•¨> and ƒÌ'p<•¨>, the above expressions becomeX=muc2/•ã
(Scalar or Pseudoscalar Theory) (76b)
Inserting these expressions in (74), H(2)ƒÀ reduces toH(2)ƒÀ=4ƒÎh2c2•èdvhdvl[g1g'1{M0M'0ƒ¢H/ƒÈ2-(MM')+(M,gradh)(M',gradh)/ƒÈ2}+g2g'2{(S,S')ƒ¢H/ƒÈ2-(S,gradh)(S',gradh)/ƒÈ 2+(T,T')(ƒ¢h/ƒÈ2-1)-(T,gradh)(T',gradh)/ƒÈ2}-g1g'2{M0(T,gradh)/ƒÈ+(M,S',gradh)/ƒÈ}-g2g'1{(S,M',gradh)/ƒÈ-(T,gradh)M'0/ƒÈ}]‡”
(Vector or Pseudovector Theory) (77a)
(Scalar or Pseudoscalar Theory) (77b) Further, by making use of the relations
4ƒÎh2c2‡”
andH(2)ƒÀ=•èdvhdvl[-f1f'1WW'+f2f'2(M,gradh)(M',gradh)/ƒÈ2+f1f'2W(M',gradh)-f2f'1(M',gradh)W'/ƒÈ]e-ƒÈ|ƒÁ<•¨>h-ƒÁ<•¨>l|/|ƒÁ<•¨>h-ƒÁ<•¨>l|-4ƒÎf2f'2/ƒÈ2•çdvM0M'0. (Scalar or Pseudoscalar Theory) (78b)
From (70) and (78), we obtain the total matrix elements for each of the four types of the meson theory:
orH(2)ƒÀ=4ƒÎh2c2•èdvhdvl[-f1f'1WW'+f2f'2{M0M'0(ƒ¢h/ƒÈ2-1)+(M,gradh)(M',gradh)/ƒÈ2}+f1f'2W(M',gradh)/ƒÈ-f2f'1(M,gradh)W'/ƒÈ]‡”
where we have ignored all the terms involving ƒÏ(h)1 or ƒÏ(h)2, because the
velocity of the heavy particle in the nucleus may be considered as
small compared with that of light.
The expression (79a) is identical with the result obtained in IV
from the unquantized theory(8).
Now, the wave lengths of the light particles are always large com
paredwith 1/ƒÈ, so that the integration with respect to the light parti clecan be carried out by ignoring the space dependence of the light
particle, wave functions. In this approximation (79) becomes
HƒÀ=4ƒÎ/ƒÈ2•çdv[g1g'1M0M'0+2/3g2g'2(S,S')], (Vector Theory) (80a)*HƒÀ=-4ƒÎ/ƒÈ2•çdv[2/3g1g'1(M,M)+1/3g2g'2(T,T)],(Pseudovector Theory) (80b)HƒÀ=-4ƒÎf1f'1/ƒÈ2•çdvWW', (Scalar Theory) (80c)HƒÀ=-4ƒÎf2f'2/3ƒÈ2•çdv(M,M). (Pseudoscalar Theory) (80d)
(8)H.A. Bethe and L.W. Nordheim, Phys. Rev. 57 (1940), 993.(*) The factor 2/3 in the second term was omitted by mistake in IV. 1941] On Yukawa's Theory of the Beta-Disintegration. 305
These expressions are identical with the original Fermi interactions,
the only differences being that, the constants are now determined by
the mesic charges of the heavy and light particles.
The calculation of the beta-decay probability proceeds then on the
same line as in Fermi's theory. Neglecting, the Coulomb force be
tweenthe nucleus and the electron, we find for the probability per
unit time that the electron with the energy between ƒÃmc3 and (ƒÃ+dƒÃ)mc2
is emittedW(ƒÃ)dƒÃ={|M1|2/T1+1/3|M2|2/T2}ƒÃ•ã<ƒÃ2-1>(ƒÃ0-ƒÃ)2dƒÃ, (Vector Theory) (81a)W(ƒÃ)dƒÃ=1/3|M2|2/T'2ƒÃ•ã<ƒÃ3-1>(ƒÃ0-ƒÃ)2dƒÃ, (Pseudovector Theory) (81b)W(ƒÃ)dƒÃ=|M1|/T'12ƒÃ•ã<ƒÃ2-1>(ƒÃ0-ƒÃ)2dƒÃ, (Scalar Theory) (81c)W(ƒÃ)dƒÃ=1/3|M<•¨>2|2/T''2ƒÃ•ã<ƒÃ2-1>(ƒÃ0-ƒÃ)2dƒÃ, (Pseudoscalar Theory) (81d)with1/T1=8/ƒÎmuc2/h(m/mu)5(g1g'1/hc)2,1/T2=32/3ƒÎmuc2/h(m/mu)5(g2g'2/hc)2,1/T'2=1/3ƒÎmuc2/h(m/mu)5{32(g1g'1/hc)2+(g2g'2/hc)2},1/T'1=8/ƒÎmuc2/h(m/mu)5(f1f'1/hc)2.1/T'2=1/3ƒÎmuc2/h(m/mu)5(f2f'2/hc)2,} (82)M1=•çƒµƒ³dv,M2<•¨>=•çƒµƒÐƒ³dv,} (83)
The energy distribution (81) is exactly the same as that obtained
from Fermi's theory. The terms involving M1 give rise to transitionsof the Fermi type, whereas those involving M2<•¨> to transitions of theGamow-Teller type(9). 306 Shoichi SAKATA. [Vol. 23III
. The Lifetime of the Meson.
It was first suggested by Yuhawa that the meson can disinte
gratespontaneously into an electron and a neutrino even in vacuum. According to the theory described in I, the probability per unit time of the disintegration of a negative meson, for example, by emitting an electron in the direction within the solid angle dĦ is given
dw=2ƒÎ/h‡”|(F|H'g'|A)|2p2_??_dpldƒ¶/(2ƒÎhc)3dEF, (84)
where pl<•¨>/c denotes the momentum of the electron and where EF is the
total energy of the final state. Further, ‡” means the summations
over both spin directions of the electron and the neutrino. If the
meson is initially at rest, (18) reduces to
dw0=muc2/32ƒÎ2h4‡”|(F|H'g'|A)|2dƒ¶. (85) The matrix elements can be easily obtained from (22) or (63):(F|H'g'|A)=-•ã<4ƒÎhc/ƒÈ>(pl<•¨>|ƒÌp=0<•¨>|pl<•¨>)u(p=0<•¨>,ƒÃ=-1) (86)with(pl<•¨>)|ƒÌp-0<•¨>|pl<•¨>)=(g'2uc(pl<•¨>)ƒÏ(l)2)ƒÐ<•¨>(l)uƒË(pl<•¨>),g'1uc(pl<•¨>)ƒÏ(l)1(pl<•¨>)ƒÐ<•¨>(l)uƒË(pl<•¨>)),(Vector Theory) (87a)
(pl<•¨>)|ƒÌp-0<•¨>|pl<•¨>)=(-g'2uc(pl<•¨>)ƒÏ(l)3)ƒÐ<•¨>(l)uƒË(pl<•¨>),g'1uc(pl<•¨>)ƒÐ<•¨>(l)uƒË(pl<•¨>)),(Pseudovector Theory) (87b)(pl<•¨>)|ƒÌp-0<•¨>|pl<•¨>)=(-f'1uc(pl<•¨>)ƒÏ(l)3)uƒË(pl<•¨>),f'2uc(pl<•¨>)uƒË(pl<•¨>)),(Scalar Theory) (87c)(pl<•¨>)|ƒÌp_??_0<•¨>|pl<•¨>)=(-f'1uc(pl<•¨>)ƒÏ(l)2uƒË(pl<•¨>),f'2uc(pl<•¨>)ƒÏ(l)1uƒË(pl<•¨>)).(Pseudoscalar Theory) (87d) Here, uc and uƒË denote the Dirac amplitudes of the electron and the
neutrino, respectively.
By using the explicit forms (28) or (59) of u , and with the equa tions(87), (86) becomes(F|H'g'|A)=-•ã<2ƒÎhc/ƒÈ>uc(pl<•¨>){g'1ƒÏ(l)1(ƒÐ<•¨>(l)e<•¨>+ig'2ƒÏ(l)2<•¨>(ƒÐ<•¨>(l)e<•¨>)}uƒË(pl<•¨>),(Vector Theory) (88a)(F|H'g'|A)=-•ã<2ƒÎhc/ƒÈ>uc(pl<•¨>){g'1(ƒÐ<•¨>(l)e<•¨>-ig'2ƒÏ(l)3<•¨>(ƒÐ<•¨>(l)e<•¨>)}uƒË(pl<•¨>),(Pseudovector Theory) (88b)(F|H'g'|A)=-•ã<2ƒÎhc/ƒÈ>uc(pl<•¨>){if'1ƒÏ(l)3(l)-f'2}uƒË(pl<•¨>), 1941] On. Yukawa's Theory of the Beta-Disintegration . 307
(Scalar Theory) (88c)F|H'g'|A)=-•ã<2ƒÎhc/ƒÈ>ue(pl<•¨>){if'1ƒÏ(l)2(l)-f'2ƒÏ(l)1}uƒË(pl<•¨>),
(Pseudosealar Theory) (88d) where the unit vector e<•¨>denotes the. direction of the polarization of the vector meson. By inserting these expressions in (85), and by performing the spin summations of the light particles with the help of the spur technique and the integration with respect to the, direction of the emission, we find the reciprocal lifetime for a meson at rest:1/ƒÑ0=muc2/2h{2/3g'31/hc+1/3g'22/hc}, (Vector Meson) (89a)
1/ƒÑ0=muc2/2h{2/3g'2_??_/hc+1/3g'22/hc}, (Pseudovector Meson) (89b)1/ƒÑ0=muc2/2h{f'21/hc+f'22/hc(m/mu)2}, (Scalar Meson) (89c)1/ƒÑ0=muc2/2h{f'1/•ã
meson moving with a velocity ƒÀc is then given by
ƒÑ=ƒÑ0/•ã<1-ƒÀ2>.(90)
It was first stressed by Nordheim(4) that in the vector theory there is a serious quantitative discrepancy between the calculated and ex perimentalvalues of the lifetime of the meson. This can be seen as follows. Inserting (82) in (89a), we obtain the relation
1/ƒÑ0=ƒÎ/16(mu/m)5{2/3(hc/g21)1/T1+1/4(hc/g22)1/T2}.(91)
The constants T1 and T2 can be determined by comparing the beta
decayformula (81a) with the lifetimes of the light elements. If we
assume that the matrix elements M1 and M2<•¨> have their maximum
possible values, these constants come out to be of the order of 103sec. Further, the values of the constants g1 and g2 determined from the
magnitude of the nuclear forces are g21/hc-0.1 and g22/hc-0.1 Using
these values of the coustants and with the value mu-170m, we find
from (91)
ƒÑ0-3.7•~10-6sec. On the other hand, measurements of the meson decay in cosmic rays
yield a lifetime of about 1-5•~10-6sec. Thus, the theoretical estimate 308 Shoichi SAKATA. [Vol. 33 is too short by a factor 10-2 in comparison with the experimental value. In the pseudovector or scalar meson theory, we meet also the same difficulty. It is, however, to be noticeable that such a difficulty disappears in the pseudoscalar meson theory, since there is no relation like (91). In this case, we can determine the value of f'1 from the meson decay and that of f2' from the beta-decay, separately. In spite of this advantage, we mast, however, abandon this possibility, since the nuclear forces derived from this theory do not agree with the experiencet. In order to avoid this discrepancy, Moller, Rosenfeld and Rozen thal(10) assumed that there exist two kinds of mesons and that while one of them has a lifetime of the order of 10-6sec in accord with ex periments,the lifetime of this other is so short that it can explain the sufficiently rapid nuclear beta-decay*. On the other hand, the introduction of two types of meson has also be required from the discussion of the problem of nuclear forces made by Moller and Rosenfeld(11). According to them, a satisfactory field theory of nuclear forces must be such as not to give rise to any static potential of the dipole type. This condition together with the requirement that the forces between a proton and a neutron should lead to the correct positions of the 3S ground level and excited 1S level of the deuteron, can not be satisfied by any theory involving only one of the four possible types of fields. However, if we try to compose two types of mesons, there is one possibility, viz. a mixture of a vector meson field and a pseudoscalar meson field, satisfying the condition(12)f22=g22. (92)
Now, it, is to be remarked that their arguments concerning the nuclear forces can also be extended to the static interaction between a heavy particle and a light particle or between two light particles . From the same requirements as before, we obtain the similar condi tionf'22=g'22. (93)
In spite of the disturbance brought about by this relation, we may still have a sufficiently long lifetime for the pseudoscalar meson, since it depends only upon the value of f'1. This result seems to be in farour of the assumption made by Moller and others.
(10) C. Moller, L. Rosenfeld and S. Rozenthal, Nature, 144 (1939),761. (*) An alternative way of removing this discrepancy proposed by the presenta uthor (Phys. Rev. 58 (1940),576) will be discussed in a separate paper. (11) Moller and Rosenfeld, Det Kgl. Danske Vidensk. Sclsk. XVII, 8 (1940). (12) Moller, Phys. Rev. 58 (1040),1118. 1941] Conduction Electrons and Thermal Expansion . 309
In conclusion, the author wishes to express his deep gratitude to Professor H. Yukawa for his kind interest in this work Institute of Theoretical Physics , Kyoto Imperial University.
(Received Feb 12, 1941.)
(_??_) [Addedin Proof: Recently it has been shown by Rarita and Sclwinger (Phys. Rev. 59 (1941),436) that the nuclear forces derived from the pseudoscalar meson theory agree in sign and spin dependence with the sign and magnitude of the singlet triplet difference and quadripole moment of the deuteron system, if we take the influence of the non-central force into account correctly. In this connection it may also be remark edthat the experimentally observed burst frequency of the meson is in good agreement with the calculation based on the pseudoscalar theory (R, F. Christy and S. Kusaka, Phys. Rev. 59 (1941). 436; See also J. R Oppenheimer, Phys. Rev. 59 (1941),462)]
Contribution, of the Conduction Electrons in a Metal to the Thermal Expansion.
By Ziro (ReadMIKURA. Nov. 2. 1940.)
•ôNH•ô Abstract•ôNS•ô.
The approximate proportionality between the thermal-expansion coefficient and the atomic heat of a metal in the moderate temperature region is well-known. However, it seems that this relation should be changed at those extremely low temperatures, where the contribution of the conduction electrons to the atomic heat is relatively large. It is the purpose of this paper to construct a formula for thermal expansion to cover this temperature region. The result is that in this region the thermal-expansion coefficient is no longer proportional to the atomic heat, and that it consists of two parts, corresponding to the Debye and the electronic terms of the latter respectively. The electronic term of the thermal-expansion coefficient is generally not proportional to that of the atomic heat. The former, however, as well as the latter, is pro
portionalto the absolute temperature at low temperatures. From the new formula a possible explanation of the small expansibility of •gInvar•h has been suggested.
ƒÌ 1. Introduction.
The well-known theoretical relation between the thermal-expansion