PHYSICAL REVIEW LETTERS
VoLUME 1$, NUMBER 16
19 OcTQBER 1964
BROKEN SYMMETRIES AND THE MASSES OF GAUGE BOSONS
Tait Institute of Mathematical
Peter W. Higgs
Physics, University of Edinburgh,
(Received 31 August 1964)
In a recent note' it was shown that the Goldstone theorem, that Lorentz-covaria. nt field
theories in which spontaneous breakdown of
symmetry under an internal Lie group occurs
contain zero-mass particles, fails if and only if
the conserved currents associated with the internal group are coupled to gauge fields. The
'
purpose of the present note is to report that,
as a consequence of this coupling, the spin-one
quanta of some of the gauge fields acquire mass;
the longitudinal degrees of freedom of these particles (which would be absent if their mass were
zero) go over into the Goldstone bosons when the
coupling tends to zero. This phenomenon is just
the relativistic analog of the plasmon phenomenon to which Anderson' has drawn attention:
that the scalar zero-mass excitations of a superconducting neutral Fermi gas become longitudinal plasmon modes of finite mass when the gas
is charged.
The simplest theory which exhibits this behavior is a gauge-invariant version of a model
used by Goldstone' himself: Two real' scalar
fields yy, and a real vector field A interact
through the Lagrangian density
L
=-&(&v
-@'7v 2 )
~
V(rp 2 + y 2 ) -P'
1
2
P, v
JL(,
where
V
p 1 =~ p 1 -eA
jL(,
p,
p2 +eA
PV
2'
{p1'
=8 A -BA
P,
e is a dimensionless coupling constant, and the
metric is taken as -+++. I. is invariant under
simultaneous gauge transformations of the first
kind on y, + iy, and of the second kind on A
Let us suppose that V'(cpa') = 0, V"(&p, ') 0; then
spontaneous breakdown of U(1) symmetry occurs.
Consider the equations [derived from (1) by
treating ~yayand A & as small quantities]
governing the propagation of small oscillations
508
Edinburgh,
about the "vacuum" solution
s "(s
Scotland
y, (x) =0, y, (x) = y, :
(np 1 )-ep 0A ) =0,
(2a)
(&'-4e, 'V" (y, ')f(&y, ) = 0,
s
V
r"'=eq (s"(c,p, )
1
0
(2b)
ep 0A -t.
(2c)
p,
Equation (2b) describes waves whose quanta have
(bare) mass 2po(V"(yo'))'"; Eqs. (2a) and (2c)
may be transformed, by the introduction of new
var iables
fl
G
IL(. V
=A
p.
-(ey 0) '8
=8 B
p.
p,
(n, (p
-BBp, =F
V
1'),
LL(V
into the form
8
B =0,
8
v
t" Pv +e 2 y 2 8
0
=0.
(4)
Equation (4) describes vector waves whose quanta
have (bare) mass ey, . In the absence of the gauge
field coupling (e =0) the situation is quite different: Equations (2a) and (2c) describe zero-mass
scalar and vector bosons, respectively. In passing, we note that the right-hand side of (2c) is
just the linear approximation to the conserved
current: It is linear in the vector potential,
gauge invariance being maintained by the presence of the gradient term. '
When one considers theoretical models in
which spontaneous breakdown of symmetry under
a semisimple group occurs, one encounters a
variety of possible situations corresponding to
the various distinct irreducible representations
to which the scalar fields may belong; the gauge
field always belongs to the adjoint representation. ' The model of the most immediate interest is that in which the scalar fields form an
octet under SU(3): Here one finds the possibility of two nonvanishing vacuum expectation values, which may be chosen to be the two Y=0,
I3=0 members of the octet. There are two
massive scalar bosons with just these quantum
numbers; the remaining six components of the
scalar octet combine with the corresponding
components of the gauge-field octet to describe
VOLUME
13, NUMBER 16
PHYSI CAL RE VIE%' LETTERS
In the present note the model is discussed mainly in
classical terms; nothing is proved about the quantized
theory. It should be understood, therefore, that the
conclusions which are presented concerning the masses
of particles are conjectures based on the quantization
of linearized classical field equations. However, es-
massive vector bosons. There are two I= ~
vector doublets, degenerate in mass between
F =+1 but with an electromagnetic mass splitting between I, =+&, and the I, =+1 components
of a V =0, I=1 triplet whose mass is entirely
electromagnetic. The two Y =0, I=O gauge
fields remain massless: This is associated
with the residual unbroken symmetry under the
Abelian group generated by Y and I, . It may be
expected that when a. further mechanism (presumably related to the weak interactions) is introduced in order to break Y conservation, one
of these gauge fields will acquire mass, leaving
the photon as the only massless vector particle.
A detailed discussion of these questions will be
presented elsewhere.
It is worth noting that an essential feature of
the type of theory which has been described in
this note is the prediction of incomplete multiplets of scalar and vector bosons. It is to be
expected that this feature will appear also in
scalar
theories in which the symmetry-breaking
fields are not elementary dynamic variables but
bilinear combinations of Fermi fields. '
sentially the same conclusions have been reached independently by F. Englert and R. Brout, Phys. Rev.
Letters 13, 321 (1964): These authors discuss the
same model quantum mechanically in lowest order
perturbation theory about the self-consistent vacuum.
~In the theory of superconductivity
such a term arises
from collective excitations of the Fermi gas.
6See, for example, S. L. Glashow and M. Gell-Mann,
Ann. Phys. {N.Y. ) 15, 437 {1961).
These are just the parameters which, if the scalar
octet interacts with baryons and mesons, lead to the
Gell-Mann-Okubo and electromagnetic mass splittings:
See S. Coleman and S. L. Glashow, Phys. Rev. 134,
B671 (1964) .
Tentative proposals that incomplete SU(3) octets of
scalar particles exist have been made by a number of
people. Such a role, as an isolated Y = ~1, I =~ state,
was proposed for the K meson (725 MeV) by Y. Nambu
and
Sakurai, Phys. Rev. Letters 11, 42 (1963).
More recently the possibility that the 0 meson (385
MeV) may be the Y=I=O member of an incomplete
octet has been considered by L. M. Brown, Phys. Rev.
Letters 13, 42 (1964).
the scalar fields
In the theory of superconductivity
are associated with fermion pairs; the doubly charged
excitation responsible for the quantization of magnetic flux is then the surviving member of a U(1) doub-
J. J.
P. W.
Higgs, to be published.
Nuovo Cimento 19, 154 (1961);
Goldstone, A. Salam, and S. %einberg, Phys. Rev.
127, 965 (1962).
P. W. Anderson, Phys. Rev. 130, 439 (1963).
J.
19 OcTOBER 1964
J. Goldstone,
let.
SPLITTING OF THE 70-PLET OF SU(6)
Mirza A. Baqi Bdg
The Rockefeller Institute,
New
York, New York
and
Virendra Singh*
Institute for Advanced Study, Princeton, New Jersey
(Received 18 September 1964)
In a previous note, ' hereafter called I, we
proposed an expression for the mass operator
responsible for lifting the degeneracies of spinunitary spin supermultiplets
[Eq. (31)-Ij. The
purpose of the present note is to apply this expression to the 70-dimensional representation of
1.
SU(6).
nances.
35@56 = 56' 707001134,
it follows that 70 is the natural candidate for accommodating the higher meson-baryon reso-
since the SU(3) CgISU(2)
70= (1, 2)+(8, 2)+(10, 2)+ (8, 4),
(2)
we may assume that partial occupancy of the 70
representation
has already been established
octet' (32) . Recent exappear to indicate that some (';)
states may also be at hand. ' With six masses at
one's disposal, our formulas can predict the
masses of all the other occupants of 70 and also
provide a consistency check on the input. Our
discussion of the 70 representation thus appears
to be of immediate physical interest.
509
through
The importance of the 70-dimensional representation has already been underlined by Pais. '
Since
Furthermore,
content is
the so-called y
periments
