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#smallcaps[*Volume 10, Number 12*]
#h(1fr)
#smallcaps[*Physical Review Letters*]
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#smallcaps[*15 June 1963*]
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UNITARY SYMMETRY AND LEPTONIC DECAYS
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Nicola Cabibbo
CERN, Geneva, Switzerland
(Received 29 April 1963)
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We present here an analysis of leptonic decays based on the unitary symmetry
for strong interactions, in the version known as "eightfold way,"#super[1]
and the $V-A$ theory for weak interactions.#super[2, 3] Our basic
assumptions on $J_mu$, the weak current of strong
interacting particles, are as follows:
(1) #underline[$J_mu$ transforms according to the eightfold
representation of $"SU"_3$]. This means that we neglect
currents with $Delta S =-Delta Q$, or $Delta I="3/2"$,
which should belong to other representations. This limits the scope of the
analysis, and we are not able to treat the complex of $K^0$
leptonic decays, or $Sigma^+ arrow n + e^+ + nu$
in which $Delta S =-Delta Q$ currents play a role. For the
other processes we make the hypothesis that the main contributions come
from that part of $J_mu$ which is in the eightfold representation.
(2) #underline[The vector part of $J_mu$ is in the same octet
as the electromagnetic current]. The vector contribution can then be
deduced from the electromagnetic properties of strong interacting
particles. For $Delta S=0$, this assumption is equivalent
to vector-current conservation.#super[2]
Together with the octet of vector currents, $j_mu$, we
assume an octet of axial currents, $g_mu$. In each
of these octets we have a current with $Delta S = 0$,
$Delta Q = 1$ $j_mu^{(0)}$ and
$g_mu^{(0)}$, and a current with $Delta S = Delta Q = 1$
$j_mu^{(1)}$ and $g_mu^{(1)}$. Their
isospin selection rules are, respectively, $Delta I = 1$
and $Delta I = "1/2"$.
From our first assumption we then get
$ J_mu = a (j_mu^{(0)} + g_mu^{(0)}) + b (j_mu^{(1)} + g_mu^{(1)}). $
A restriction $a = b = 1$ would #underline[not]
ensure universality in the usual sense (equal coupling
for all currents), because if $J_mu$
[as given in Eq. (1)] is coupled, we can build a current,
$b (j_mu^{(0)} + g_mu^{(0)}) - a (j_mu^{(1)} + g_mu^{(1)})$,
which is not coupled. We want, however, to keep a weaker
form of universality, by requiring the following:
(3) #underline[$J_mu$ has "unit length,"
i.e. , $a^2 + b^2 = 1$].
We then rewrite $J_mu$ as#super[4]
$ J_mu = cos theta (j_mu^{(0)} + g_mu^{(0)}) +
sin theta (j_mu^{(1)} + g_mu^{(1)}), $
where $tan theta = "b / a"$.
Since $J_mu$, as well as the baryons and
the pseudoscalar mesons, belongs to the octet
representation of $"SU"_3$, we have relations
(in which $theta$ enters as a parameter)
between processes with $Delta S = 0$
and processes with $Delta S = 1$.
To determine $theta$, let us compare
the rates for $K^+ arrow mu^+ + nu$
and $pi^+ arrow mu^+ + nu$; we find
$ Gamma (K^+ mu nu)"/"Gamma (pi^+ mu nu) \
= tan^2 theta M_K (1 - M_mu^2 "/" M_K^2)^2"/" M_pi (1 -
M_mu^2 "/" M_pi^2)^2. $
From the experimental data, we then get#super[5, 6]
$ theta = 0.257. $
For an independent determination of $theta$,
let us consider $K^+ arrow pi^0 + e^+ + nu$.
The matrix element for this process can be connected to that for
$pi^+ arrow pi^0 + e^+ + nu$, known from
the conserved vector-current hypothesis (2nd assumption). From the
rate#super[6] for $K^+ arrow pi^0 + e^+ + nu$, we get
$ theta = 0.26. $
The two determinations coincide within experimental errors;
in the following we use $theta = 0.26$.
We go now to the leptonic decays of the baryons, of the type
$A arrow B + e + nu$. The matrix element
of any member of an octet of currents among two
baryon states (also members of octets) can be expressed in terms
of two reduced matrix elements#super[7]
$ angle.l A | j_mu^{(i)} + g_mu^{(i)} | B angle.r =
"if"_"ABi" O_mu + d_"ABi" E_mu; $
the $f$'s and $d$'s are coefficients
defined in Gell-Mann's paper.#super[1, 7] It is
sufficient to consider only allowed contributions and write
$ O_mu , E_mu = F^"O, E"gamma_mu + H^"O, E"gamma_mu gamma_5. $
From the connection with the electromagnetic current we get the
vector coefficients: $F^O = 1$, $F^E = 0$;
from neutron decay we get
$ H^O + H^E = 1.25. $
We remain with one parameter which can be determined from
the rate for $Sigma^- arrow Lambda + e^- + overline(nu)$.
The relevant matrix element for this is
$ cos theta angle.l Sigma^- |
j_mu^{(0)} + g_mu^{(0)} | Lambda angle.r \
= cos theta (2/3)^"1/2" E_mu =
(2/3)^"1/2" cos theta H^E gamma_mu gamma_5 . $
Taking the branching ratio for this mode to be
$0.9 times 10^"-4"$,#super[8] we get
$ H^E = plus.minus 0.95. $
The negative solution can be discarded because it produces
a large branching ratio for $Sigma^- arrow n + e^- + overline(nu)$,
of the order of $1\%$. The positive solution
($H^E = 0.95$, $H^O = 0.30$) is
good, because it produces a cancellation of the axial
contribution to this process. This explains the experimental
result that this mode is more depressed than the
$Lambda arrow p + e^- + overline(nu)$
in respect to the predictions of Feynman and Gell-Mann.#super[2].
In Table I, we give a summary of our predictions for the electron
modes with $Delta S = 1$. The branching ratios for
$Lambda arrow p + e^- + overline(nu)$ and
$Sigma^- arrow n + e^- + overline(nu)$ are
in good agreement with experimental data.#super[9]
As a final remark, the vector-coupling constant for $beta$
decay is not $G cos theta$. This gives a correction
of $6.6\%$ to the $f$$t$ value of Fermi
transitions, in the right direction to eliminate the discrepancy
between $O^14$ and muon lifetimes.
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$Lambda arrow p "+" e^- "+" overline(nu)$, $1.4#hide[0]\%$, $0.75times 10^"-3"$, $V - 0.72 A$,
$Sigma^- arrow n "+" e^- "+" overline(nu)$, $5.1#hide[0]\%$, $1.9 #hide[0]times 10^"-3"$, $V + 0.65 A$,
$Xi^- arrow Lambda "+" e^- "+" overline(nu)$, $1.4#hide[0]\%$, $0.35times 10^"-3"$, $V + 0.02 A$,
$Xi^- arrow Sigma^0 "+" e^- "+" overline(nu)$, $1.14\%$, $0.07times 10^"-3"$, $V - 1.25 A$,
$Xi^0 arrow Sigma^+ "+" e^- "+" overline(nu)$, $0.28\%$, $0.26times 10^"-3"$, $V - 1.25 A$,
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The correction is, however, too large, leaving about $2%$ to be explained.
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#super[1]M. Gell-Mann, California Institute of Technology Report CTSL-20, 1961
(unpublished); Y. Ne'eman, Nucl. Phys. 26, 222 (1961).
#super[2]R. P. Feynman and M. Gell-Mann, Phys. Rev. 109, 193 (1958)
#super[2]R. E. Marshak and E. C. G. Sudarshan, #underline[Proceedings of the
Padua-Venice Conference on Mesons and Recently Discovered
Particles, September, 1957] (Società Italiana di Fisica, Padua-Venice, 1958);
Phys. Rev. 109, 1860 (1958).
#super[4]Similar considerations are forwarded in M. Gell-Mann and M. Lévy,
Nuovo Cimento 16, 705 (1958).
#super[5]The lifetime from W. H. Barkas and A. H. Rosenfeld,
#underline[Proceedings of the Tenth Annual International Rochester Conference
on High-Energy Physics, 1960] (Interscience Publishers, Inc., New York, 1960),
p.878. The branching ratio for $K^+ arrow mu^+ + nu$
is taken as $57.4 \%$. W. Becker, M. Goldberg, E. Hart, J. Leitner,
and S. Lichtman (to be published).
#super[6]B. P. Roe, D. Sinclair, J. L. Brown, D. A. Glaser, J. A. Kadyk,
and G. H. Trilling, Phys. Rev. Letters 7, 346 (1961). These authors give
the branching ratio for $K^+ arrow mu^+ + nu$ as
$64\%$, from which $theta=0.269$. Also
this value agrees with that from $K^+ arrow pi^0 + e^+ + nu$
within experimental errors.
#super[7]N. Cabibbo and R. Gatto, Nuovo Cimento 21, 872 (1961). Our notation for the currents is differ-
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ent from the one used in this
reference and by Gell-Mann; the connection is
$j_mu^(0) = j_mu^1 + i j_mu^2$, $j_mu^(1) = j_mu^4 + i j_mu^5$.
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#super[8]W. Willis et al. reported at the Washington meeting
of the American Physical Society, 1963 [W. Willis et al. ,
Bull. Am. Phys. Soc. 8, 349 (1963] this branching ratio as
$(0.9^"+0.5"_"-0.4") times 10^"-4"$.
If it is allowed to vary between these limits, our predictions
for the $Sigma^- arrow n e^- overline(nu)$
varies between $0.8 times 10^"-3"$ and
$4 times 10^"-3"$, and that for
$Lambda^0 arrow p e^- overline(nu)$ between
$1.05 times 10^"-3"$ and $0.56 times 10^"-3"$.
I am grateful to the members of this group for prepublication
communication of their results.
#super[9]R. P. Ely, G. Gidal, L. Oswald, W. Singleton, W. M. Powell,
F. W. Bullock, G. E. Kalmus, C. Henderson, and R. F. Stannard [#underline[Proceedings
of the International Conference on High-Energy Nuclear Physics, Geneva, 1962]
(CERN Scientific Information Service, Geneva, Switzerland, 1962), p. 445]
give the branching ratio for $Lambda arrow p + e^- + overline(nu)$
as $(0.85 \pm 0.3) times 10^"-3"$ while that for
$Sigma^- arrow n + e^- + overline(nu)$ is given
(see preceding reference) as $(1.9 plus.minus 0.9) times 10^"-3"$.
#super[10]R. P. Feynman, #underline[Proceedings of the Tenth Annual
International Rochester Conference on High-Energy Physics, 1960]
(Interscience Publishers, Inc. , New York, 1960), p. 501. Recent
measurements of the muon lifetime have slightly increased the discrepancy.
We think that more information will be needed to decide whether
our 3rd assumption can be maintained.