HUSH YI N YI L MAZ
Technology (1954) under the constant encouragement
of Professor George J. Yevick (1954). The part concerning cosmology was largely developed
at the
National Research Council of Canada, Ottawa (1956).
The author would like to express his appreciation to
Professor Philip M. Morse of the Massachusetts
Institute of Technology, to Dr. G. Herzberg and
Dr. T.-Y. Wu of the National Research Council of
PHYSICAL REVIEW
VOLUM E
Possible Experimental
111,
NUM
BER
SEPTEM BER 1, 1958
5
Test of Universal Fermi Interaction
R.
Istjtnto di Fisica e Scnola di Perfezionamento
Canada for the support and much-needed encouragement of this work in its earlier stages. Warm gratitute
is especially due to Dr. L. S. Sheingold, Dr. J. T.
Thomas, and Mr. A. S. Gutman of Sylvania without
whose continued
the
support and encouragement
investigation could not have been carried to its present
stage. The author wishes to thank S. Schneider for many
valuable suggestions and illuminating discussions.
GATTO
in Fzsica Nncleare, Istitnto Nazionale di Fisica Nncleare, Sezione di Rema, Italia
(Received April 15, 1958)
E
E.
The decay modes
particles are discussed with the aim of deriving,
a and E» of charged and neutral
in the Feynman-Gell-Mann-Marshak-Sudarshan
theory, possible experimental tests of the hypothesis of
universal Fermi interaction, which is already apparently contradicted by the present data on the ratio of
&~e+v to m —+tM, +v. Measurement of the E3 spectra would already provide a test of the hypothesis, and
measurements of the polarizations would give further con6rmation. Unique forms of the spectra of the
charged leptons are predicted on the basis of the universality hypothesis and of particular assumptions.
INTROBUCTIOlV
EYNMAN and Gell-Mann' and, independently,
Marshak and Sudarshan' have recently proposed
a theory of the weak interactions (to which we shall
briefly refer to as the FGMS theory) based upon the
assumption that the different spinor fields P are weakly
coupled only in the pro&ection -', (1+ps)f. This theory
seems to account successfully for most of the established
experimental evidence on weak interactions. The total
weak interaction is assumed to arise from the coupling
of a current Jq with itself; J), is the sum of bilinear
covariants
pairs of
yb (1+ps)ll b$ over certain
fermions a, b that satisfy particular requirements Lfor
instance, (ct, b) must be a single charged pair). The current Jb will in particular contain a part t lf „yb(1+ye)lt,
+tlf„yb(1+Tb)$„$ thus implying that tt and e have
exactly the same weak interactions (conservation of
leptons requires tt and e to be both particles).
and e have no strong couplings.
Apparently both
Under such conditions, from the hypothesis of minimal
electromagnetic interaction, p and e would also have
exactly the same electromagnetic couplings, and the
notion of relative parity between p and e would lose
any meaning. Measurements of the magnetic moment
of the muon' do not give evidence so far for a complicated structure of the muon, as would be expected if
the muon possessed strong interactions. The only
~
g
)
LM
' R. P. Feynman
and M. Gell-Mann, Phys. Rev. 109, 193 (1958).
-'
E. C. G. Sudarshan and R. E. Marshak, Proceedings of PaduaVenice Conference on Mesons and Eerily Discovered Particles,
September, 1957 LSuppl. Nuovo cimento (to be published)
3 CofFin, Garwin, Penman, Lederman,
g, nd Sachs, Phys. Rcv.
j.
109, 973 (1958).
di6'erence between the two particles would then be due
to the remarkably large difference of their masses.
Such a situation seems rather peculiar but, if de6nitely
established, may turn out to be very suggestive.
On the other hand, the only available experimental
evidence, that is directly related to the problem, is the
experimental upper limit for the ratio of rr-me+ v to
~-+tc+v. Anderson and Lattes find only a 1% probability that this ratio could be greater than 2. 1&(10 '.4
The value predicted by the FGMS theory is 13.6X10 ',
as can be shown independently of perturbation theory
for the strong interactions. ' The discrepancy may
indicate either an intrinsic difference in the interactions
of p and e or a more complicated structure of the weak
universal interaction. '
We want here to examine the possibility of an
independent test of the hypothesis of identical interaction of p, and of e through a study of the decay
modes E &ts+v+rr and E +e+v+rr enamel—
y: E+—:tt"
Ers~ts+
or
e+)+v+zr
~tt
or e+)+v+zr,
(or e )
(or
—
4 H. L. Anderson and C. M. G. Lattes, Nuovo cimento 6, 1356
(1957).
z M. Ruderman and R. Finkelstein, Phys. Rev. 76, 1458 (1949).
z It was proposed
LR. Gatto, Nuclear Phys. 5, 530 (1958)
that departures from locality, as introduced by Lee and Yang
for y decay PT. D. Lee and C. N. Yang, Phys. Rev. 108, 1611
(1957)g could account for the z~e+v to z~n+v ratio. However
j
the nonlocality required in this case would no longer be compatible with a form of the weak interaction as a coupling of the
current Jg with itself, even if such coupling is propagated through
some 6nite space-time distance. (This model would instead be
sufFicient to explain in the FGMS theory the deviations of p
from 4s, provided one removes from the theory the hypothesis
of the vector coupling. )
of nonrenormalizability
UN
IVERSAL FERMI INTERACTION
Kr, ' is the long-lived Ks which we assume to
an
be
eigenstate of CP
Since many papers have already appeared on the E3
decay modes, ' we shall limit our discussion only to
those points that are relevant to the problem of the
universal interaction. In particular we shall not be
concerned with the interesting possibility of distinguishing between the various possible decay couplings
from a study of the final spectra, as already proposed
and investigated in detail by many authors. ' We are
mainly concerned with the possibility of testing, by a
diferent experiment, the hypothesis of universal Fermi
interaction in the FGMS theory; alternatively our
results can also be used to test the assumptions of the
FGMS theory, once the universality hypothesis is
admitted.
+v+s+;
j.
In the FGMS theory the only invariant that can be
)
constructed from (1+ps) and (1+ps)P is [/P(1+ps)
where P, =cP &n&+c'P, &x& with P &n& the total fourmomentum of l and v, and E', &~' the E four-momentum;
c and c' are functions of (P&~&)' and (P& P' &) From
the Dirac equation and four-momentum conservation
+l+v+s. in
we can write the matrix element for K—
the form
&
&
&
&
P&x&)'M
(
'(m&X[t(1+ps)v]
+sY[fP& &(1+vs)~3, (1)
with X and Y unknown functions of (P& P&x&)IM,
with E& ) the m four-momentum,
M the
mass, and
vs~ the mass of the charged lepton /. From invariance
under time reversal (that is implicit in the FGMS
theory) X and Y have the same relative phase and
&
E
can be taken as real.
The intrinsic specification of the final state requires
two kinematic parameters for which we choose the
energies E and E~ of the pion and of the charged lepton
l, respectively, and also a knowledge of the state of
polarization of l. We thus write the final distribution in
the E rest system in the form
Wi(EEii () = W&&'& (EE&)+ (( n&) Wi&'&(EE&)
+(( n) W&"&(EEi),
—Wr&s&/W&&s)
From the form (1) of the matrix element, we
W&(EE&~()dEdE& gives the probability of the
decay configuration for which the pion energy is between E and E+dE, the energy of the charged lepton
between Ei and E&+dE&, and the spin of l, measured
in the L rest system, is in the direction specified by the
where
'A. Pais and S. B. Treirnan, Phys. Rev. 105, 1616 (1957);
Furuichi, Kodama, Ogawa, Sugawara, Wakasa, and Yonezawa,
Progr. Theoret. Phys. Japan 17, 89 (1957); J. Werle, Nuclear
Phys. 4, 171 (1957}; S. W. McDowell, Nuovo cimento 6, 1445
(1957); Furuichi, Sawada, and Yonezawa, Nuovo cimento 6,
1416 (1957); L. Okun, Nuclear Phys. 5, 455 (1958); J. J. Sakurai,
Phys. Rev. 109, 980 (1958).
'We denote the Dirac spinors with the same symbol as the
particles; / is the emitted charged lepton, either g or e. We neglect
electromagnetic corrections.
f'&nd
W&(s& (EE&)
= mpX (E)'(a' —p' —m p)
+ Y(E)'M'[ —(LV
W
—
E&)—
4m p X(E) Y (E)M (a
' m—
p—
p)+4E&(& E—
i) j, (3)
&1&(EE&)
pl
Ps)
(m, sX(E)s[(~+m&)s —
+2X(E) Y(E)m&M[(6' —p' —mp) —2E&(6+m&)
+ Y(E)2M2P2 ps+m 2 4E (g E )
—2m&(h —2E$)j),
j
(3')
Wr &s& (EE&)
= 2m&P[mPX(E) s —2X(E) Y (E)MEi
+ Y(E)'M'j,
(3")
where p is the magnitude of the s- 3-momentum, p7
the magnitude of the 3-momentum of /, and the argument of X and Y, (P' P&~&)IM, is, in the K rest
system, the pion energy E. 6 is defined as M K
The above expressions apply to the decay modes
leading to negatively charged electrons or muons. For
the decay modes leading to positively charged electrons
or muons, 8'"& is unchanged, W') and 8'(') take up a
negative sign.
The energy
varies between the pion rest mass, nz,
and s(M'+m~s mP)IM, and E— varies between mr
and —' (M'+m P —
m ')/M; for fixed E, E& varies between
p)'+m&'j/(~+ p);
l [(~ p)'+m&'3I(~ —
p) and sl (~+—
for fixed E&, E varies between sr[(A& —
p&)'+m ']/
—
'[(4g+p&)'+m 'j/(A&+p&), with 6&
(&& p&) and —,
&
—
8
&
,
+l.
(2)
&
&&P
Ei+mi
DECAY DISTRIBUTIONS
(P&
unit vector (; n& and n are unit vectors in the direction
of the l and w 3-momenta, respectively. The degree of
will then be given by
polarization along the direction
&&'&(EE )r+ n&P &&~&(EE )&, where &P&&'& and &Pr&
&P&(() =n
can be obtained from (2): &P&&'&= W&&'&/W&&'& &P&&~&
M-
One verifies immediately that in the limit no&=0
(which is applicable if l is the electron) &P&&" (EEi) = —1
and &P&& (EE&) = 0, as expected because of the projection
-',
(1+Vs) in front of l, implying that l is left-handed
if its rest mass can be neglected (l+ would be right&
handed).
A particular configuration is that with p=0, i.e. , in
which the pion is emitted at rest in the E rest system,
and the two leptons are emitted with equal and opposite
momenta. The matrix element for such a configuration
&Ap '
im&nz
takes the form, in the K rest system, —
X(Xd —MY)[l(1+ps)v$, with ho —M —m as can be
seen directly from (1), using the Dirac equation and
the conservation laws. In this expression X and Y
MI". Accordappear only in the combination Xhp —
&&
—
R. GATTO
1428
ingly, the ratio between the electron rate and the
muon rate is uniquely given for such a configuration
m ')' 2.&&&10-'.
m, 2)'/(Dp —
by R./R„=(m /m )'(hp —
This result is quite analogous to that pointed out by
Ruderman and Finkelstein for the vr +e+& to ~~@+»
ratio. Moreover, for this configuration the polarization
of the emitted lepton must be only longitudinal and
maximum in value in order to balance the angular
momentum taken oG by the neutrino: 6'& ——&j. and
5' =0 for p+ and e+. [The reason for the smallness of
the electron rate can in fact be understood by recalling
that in the limit no~=0 no right-handed 1 can be
emitted because of the projection -,'(1+y5).7 These
conclusions can also be verified from (3), (3'), and (3").
—
COMPARISON WITH EXPERIMENT
It
is clear in which way the hypothesis of Universal
Fermi Interaction imposes restrictions on the relative
decay probabilities for the modes E» and
3. By
integrating Eq. (3) over dEi one finds for the pion
energy distribution, 8"&(E)dE,
such an approximation to be reliable since the virtual
states contributing (which involve baryon-antibaryon
pairs) are relatively far from the energy shell. It must
however be kept in mind that such an argument is
certainly incomplete: first, because it is based on
perturbation theory for the strong couplings; second,
theory, it could be
because, even in perturbation
invalidated if cancellations occur among the various
for the diGerent baryon-antienergy denominators
baryon pairs a quite possible situation if there are
particular symmetries among the strong couplings.
Assuming X and Y to be constants we can integrate
Eq. (3) over dE to find the muon spectrum X„and the
electron spectrum Ã, (neglecting the electron mass)
—
—= x'f."'" —I+»f'" (Eu,—
(E& '&
&.
I
(M)
(E&
I
I
'l
&
I
(M)
I
E,
Wi(E)dE
where
(E) (mPX(E)'(a' p' mP)—
—4mPX(E) F (E)Ma[1 ——,'g+(E) 7
+M2P'(E)2[ (g2 p2 m 2)
pg
—
&
I
ia&I
—= F'f'" (E—i
EM)
I
I
«Mi
(+~I
(M)
I
where
+» g. (E)[1—:g,(E» —.
p g (E) ~)dE,
—
—
=
p'). Now, if all the
g~(E) (6' p'Amp)/(6'
interactions of p and of e are exactly the same, the two
functions X(E) and V(E) are the same for E„3 and for
E,3. Therefore a measurement of the E„3 rate and of
the E,3 rate for pions in a given energy interval dE
contained in both spectra can be used to determine
X(E) and Y(E). Since (4) is a bilinear expression one
finds 4 pairs of solutions [X(E),V(E) 7, of which
however only two need to be considered because the
other two diGer only by an over-all change in sign of
the total matrix element. Inserting the two solutions
into (a) one determines two possible muon spectra
(for pious of the energy E), which are the only two
allowed under the hypothesis of identical interaction
of p and of e (note that the electron spectrum is uniquely
determined in the approximation m, =0). Furthermore,
corresponding to each of the two energy spectra, 8'„&"
and 8'„(') are uniquely determined, and the complete
muon polarization (longitudinal (P„i» and along the m
(P~& &) is thus
momentum
uniquely predicted. This
polarization can be measured from a study of the
E &p—+e asymmetry. Of course, the most convenient
procedure for actually testing the equality of X(E)
and 7(E) for E'» and E,3 in Eqs. (3), (3'), and (3")
may well be different from the one we have just described.
Since no data for a complete comparison of the kind
outlined above are available so far, we shall now
discuss an approximate
comparison, based on the
assumption that the energy dependence of X(E) and
I'(E) can be neglected. One would genera, lly expect
—
+y2f
——
h+(E.
f, "'I
(E„)
—=
I
p„(— —
4 h
m„$
—
-I
(E."&M)
)
I
M
[1—lh+(E. )7,
(E~)
p~
(m
—= —
&-(E.) — 1+
f."'I&M)
(M)
M
(m&'&
p
I
—+
(M)
I
with hg(E„) = (6„'—
p„'am~')/(6
(E,
EM)
I
—4I
+4—
(Eq'(
E~
'&
),
M
2
——
1
M
)M
' —p ')
and
I
(mq'
h+(E~)
E, '
(M) (
To be definite, let us consider here the decay of the
charged E's; of course, all the subsequent discussion
also applies to neutral J&. . The functions f„"&, f„&",
f„~3', and f, &'& are shown in Fig. 1. By numerical integration we Qnd for the total decay rates [Mr(X„3)7 '
=5)&10 'X' —2X10 'XV+5X10 'I ' [Mr(& 3)7 '=6
)&10 'V'. These two equations can be used to determine
X and V. Again one finds four possible solutions. Only
two of them have to be considered, as the other two
diGer only by an overall change in sign. There is
however a condition to be satisfied in order that the
UNIVERSAL
solutions
namely,
be acceptable
FERMI INTERACTION
(X and F must be real),
I:r(E.S)/r(Eos)3
~&
Np (first solution)
-o 5
This condition is satisded by the present inaccurate
data on r(E, s) and r(E„s).' A similar condition must
of course be satisfied by neutral E's.
Inserting the tentative values for the partial decay
rates given in reference 9, we find the two independent
possibilities X=4.6&&10'(Mev sec) ', F'= 1.1X10'(Mev
0.37&&10'(Mev sec) '*,
sec) ' (first solution) and X= —
F=1.1X10'(Mev sec) i (second solution). The corresponding electron spectrum X, (which is uniquely
determined) and the two possible muon spectra are
reported in Fig. 2. The areas under the spectra are
proportional to the partial rates of the corresponding
decay mode the areas are, respectively, normalized
to 10 eMr(E. s) and 10 eMr(E„s)]. Thus Fig. 2 gives
directly the number of e and the number of p to be
found at the energy E&.
We want to stress here two points: erst, the values
that we have inserted for the E» rates are quite
preliminary' and may have rather large errors (when
better values become available one should, using the
graphs in Fig. 1 determine the spectra more accurately);
second, the spectra reported in Fig. 2 are based on the
assumption that the energy dependence of X and F
may be neglected, which cannot be rigorously justified
I
at present.
We have already examined the particular con6guration in which the m is emitted at rest in the E rest
system. For this particular configuration the ratio
between the E,3 mode and the E„3 mode must be
0 030-
0 Qe&0-
L0$0.
0.10
0.20
0.30
0.40
FIG. 2. The e spectrum N, dE, and the two possible p spectra
X„&fE of E+ decay obtained for the values (r(ft»)] '=3.26&&10&&
and
r(E„)] =3.42X10e. The areas under the spectra are
proportional to the corresponding partial decay rates.
2.6)&10 ' if the hypothesis of exactly equal interaction of p, and of e is correct.
We have still to examine the practical use of such a
result for testing the hypothesis on which it is based.
From the form (4) of the pion spectrum we can calculate
the E„3+ and the E,3+ rates for the total of all those
configurations for which the pion momentum in units
of the E mass is less than (p/M). We find:
0.20F)'(p/M)' —(3/5) (0.17X'
(E„s+ rate)=(0. 19X—
—0.19XF'+0.40F') (p/M) '+
0.20F)'X2.6X10
(0.19X—
(E,s+ rate) —
s(P/M)
+0.67 Fs(3/5) (p/M) '+
where we have neglected the energy variation of X and
F in the vicinity of p = 0. Inserting the tentative values
found above for X and I' we find that the ratio
(E,s+ rate)/(E„s+ rate) for configurations close to that
with the pion at rest, is given by 2.6&(10
1(p/M)'
for the first solution, and by 2.6)&10 '+5.6(p/M)' for
the second solution. One notices that the rapid increase
of the second term in each of the two expressions will
comparison rather diKmake this model-independent
'+1.
0.020-
cult.
CONCLUSIONS
The most direct test of the hypothesis of identical
interaction of p and e derivable from the E3 spectra is
that obtained for the configurations with the pion
emitted at rest in E rest system. The rapid variation
of the ratio of the two decay rates with the pion energy
FIG. i. The functions
spectrum:
appearing in the
dificult.
will however make this test experimentally
= (X'f "'+XYf„&'&+Y'ff&'&)dE» and in the p,e spectrum: N„dE„
f&f,&fZ,
= F f,(3)dE, . E& is the total energy of p or e and M is the E mass. In the general case, if the pion energy is measured, the
X and Y are to be determined from the normalizations j'E„&fE„
electron energy spectrum is uniquely predicted and
= L-(X. ) J'~.dE. =L-(E. )]-.
only two possible muon spectra are allowed. For each
M. Gell-Mann and A. H. Rosenfeld, in A&&&&Nal Eeofe&o of of such spectra the complete muon polarization (longiEgclear Sciemce (Annual Reviews, Inc. , Palo Alto, j.957), Vol. 7,
tudinal and along the pion direction) is uniquely
p. 457.
. r,
&&
.
R. GATTO
I430
predicted. If the assumption of a slow energy variation
of X and I' is accepted, the knowledge of the partial
decay rates is sufficient to determine uniquely the
electron spectrum and the two possible muon spectra
(and corresponding the average polarizations), without
need of observing the pion emitted. The general
requirement r(K, s) &~ ~0.5r(K„s) which follows from
the theory, is apparently satisfied by the present data.
+e+o to
If the present discrepancy with the sr —
—
be consr +tt+o ratio will still persist and eventually
6rmed by further tests of the kind examined here,
then some of the following possibilities should be
considered in detail:
The hypothesis of a universal interaction, in the
sense of strict equality of coupling constants, is not true.
The hypothesis of universality is valid but the
universal interaction, still local, has a more complicated
for instance, a
form than the simple A&U mixture
term is present which almost
small pseudoscalar
exactly cancels the contribution from 2 to electron
decay. .
The universal interaction is nonlocal, such that the
—
PHYSICAL REVIEW
VOLUME
111,
two leptons are emitted at different points Lbut always
with the projection -,'(1+ps)$. Such a nonlocality must
however extend up to very long wavelengths corresponding to a mass of 100 Mev or even less.
The universal interaction is nonlocal and furthermore
the leptons are not required to interact only through
the projection —,(1+ps) but such a requirement is only
valid in the local limit. Such a form of the interaction
has been proposed by Sirlin" and it oGers a more
redundant solution of the n.~e+o problem than the
simpler introduction of a small local pseudoscalar term.
Moreover, as pointed out by I'eynman and Gell+e+p be
Mann, the requirement that the rate of tt —
slow imposes stringent conditions.
If one wants to insist on the universal A& V form,
of
one can speculate about a possible breakdown
present electrodynamics that may oGer a possibility
for an explanation.
—
'
"
"
A. Sirlin, Phys. Rev. 111, 337 (1958).
R. P. Feynman and M. Gell-Mann (private communication).
Gatto and M. Ruderman, Nuovo cimento 8, 775 (1958);
9, 556 (1958).
44
~'
"R.
NUM
BER
SEPTEMBER 1, 1958
5
Vertex Function in Quantized Field Theories*
REINHARD
OEHME
of Physics attd the Frtrico Fermi Irtstitute for Nuclear Studies, IIrtioersity of Chicago, Chicago, Illiuois
(Received April 18, 1958)
Departmertt
is given for the vertex function F(hs, ps, (h —
An integral representation
p)4). This representation is
obtained on the basis of local commutativity and the spectral conditions. It exhibits the set of points k', p~,
—
for which F is analytic in (h P)s except for the physical cut. The limitations found in the representation
are discussed on the basis of examples obtained from perturbation theory. These examples give some insight
into the question of further analytic continuation in k' and p'.
' 'N a
previous article' we have obtained certain
analyticity properties of the vertex function on the
basis of the general axioms of field theory. However,
the information contained in these axioms has not been
completely exhausted, and especially the unitarity
condition was not used at all. It is the purpose of the
present note to give a representation for the vertex
function which exhibits the analytic properties obtained
so far, and to discuss the limitations which one encounters on the basis of examples obtained from
perturbation theory.
~
-
Let us consider first the general vertex function
G(k, p)
=,
f
'
de~
d4y
e'"'* '"'"G(x y)
(1)
*This work has been performed while the author was at the
Institute for Advanced Study, Princeton, New Jersey.
'Bremermann,
Oehme, and Taylor, Phys. Rev. 109, 2178
(1958). This paper will be referred to as BOT; it contains further
references.
where G(a, y) may be written in the form
~)(0 LLa(3), A (~) j,c(0) o)
G(~, y) =e(*)(e(y —
+e(y)&o IP (~), I:Ilb), c(o)2 IO)}. (2)
I
j
I
The quantities A, 8, C can be considered as current
operators which satisfy the spectral conditions (0 A rt)
=0 unless p„'&a', (OIBIrt)=0 unless p '&bs, and
(OICIrt)=0 unless p„'&c'. Here p with p„s)0 is the
four-vector describing the total energy and momentum
of the state Ie). Using the methods described earlier
one can show that G(k, p) is a boundary value of an
analytic function F(stssss) with st —k', ss —p', ss
= (k —p)', s; =x,+iy;; this function F is regular in the
s3 plane except for a cut y3=0, xs&c', provided the
variables s~, s2 are restricted to a certain domain D in
the space of two complex variables. The domain D is
most easily obtained if the methods of SOT are supplemented by the general representation for the causal
commutator, which has been proven by Dyson. ' Using
I
s
F. J. Dyson,
Phys. Rev. 110, 1460 (1958).
I