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leading to the Gell-Mann and Okubo relation
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(11-26)
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For the pseudo scalar octet (K, n, 1]) the analogous formula involves empirically square masses
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+ m;
(11-27)
We should allow for some mixing with a ninth 1]' meson to complete a nonet. Using Eq. (11-27) at face value, it predicts m~ = 0.320 GeV 2, the experimental value being m~ = 0.301 GeV 2 . Octet dominance applied to the resonance decuplet leads to equal-mass spacing, in good agreement with the observed data and considered as a major triumph in its prediction of the complete set of quantum numbers of the Qparticle: mp - mil = 152 MeV
m = mo
+ msY
m=:_ - mp
mg -
149 MeV
(11-28)
mE- = 139 MeV
Unitary symmetry leads to numerous relations among scattering amplitudes which are in satisfactory agreement with experiment. We shall have more to say about it when we discuss the applications of current algebra.
11-2-2 Spontaneous Symmetry Breaking
Spontaneous symmetry breaking occurs when the ground state is not invariant under the transformation group. The latter acts in a larger nonseparable Hilbert space, as was exemplified by the behavior under rotations of the infinite ferromagnet (Chap. 4). We can convince ourselves of this fact by the following heuristic remark. Let us try to compute the norm of the state Q 10), where Q is the wouldbe total charge (11-29) From translational invariance this is also equal to
d x <0 Jjo(O)Q 10)
and is clearly infinite when Q 10) =1= O. The most striking consequence analyzed by Goldstone is the appearance of massless particles when the broken symmetry is a continuous one. These states are generated by operators which would rotate the vacuum by an infinitesimal
520 QUANTUM FIELD THEORY
amount to a degenerate vacuum, since on physical grounds such a transformation does not cost any energy. To prove Goldstone's theorem we assume the existence of a conserved current and consider any operator A such that
t5a(t) == lim
v-+co
<01 [Qv(t), A] 10> =1= 0
(11-30)
That there exists such an observable just expresses the noninvariance of the vacuum. Insert a complete set of intermediate states of definite four-momentum in this relation:
t5a(t)= lim
V-+oo
d 3 x[<0Ijo(0)ln><nIAI0>e- iP ,.x-
0> <01 A In><nljo(O) 1 eiP,.X]
(11-31)
(2n t5 3 (p n) [<0 Jjo(O) In><nl A 10> e-iE.t -
<01 A In><nJjo(O) 10> eiE.t]
We have already shown in Eq. (11-10) that current conservation implies
d dt t5a(t) = 0
(11-32)
Therefore
0= I
(2n)3 t5 3(Pn)En[ <0 Jjo(O) In><nl A 10> e-iE.t
+ <01 A In><nJjo(O) 10> eiE.t]
(11-33)
It then follows from Eqs. (11-31) and (11-33) that a state In> must exist such 3 that A In><nJjo(O) =1= for which E n t5 (P n ) vanishes. This is a massless state with the same quantum numbers as jo (and A) since it is generated by this operator
10> 0
from the vacuum. These massless states are called Goldstone bosons associated to the symmetry breaking. They are indeed bosons ifjo is a boson-type operator. In a more general setting such as the recent supersymmetric theoriesjll could in fact carry half-integer spin and the associated massless particle would then be a fermion. In other words, the spin of these states bears a relation with the Lorentz transformation properties
ofjll(x).
A subtle point is that the massless states need not necessarily be observable. This remark applies to theories with an unphysical sector of unobservable states (such as quantum electrodynamics in the Gupta-Bleuler gauge) and may be relevant when we want to avoid the conclusion of the theorem. The mechanism of spontaneous symmetry breaking is characterized by its aesthetic appeal, as opposed to the ad hoc breaking prescriptions such as those encountered when discussing the octet model. It is economical in the sense that it does not introduce any new parameter. It is also theoretically advantageous since it preserves renormalizability properties. As a counterpart the appearance of massless particles might be an unpleasant feature in certain applications. We
SYMMETRIES
shall encounter in the next chapter an elegant way to dispose of them using Higgs' mechanism.
As an extreme example of spontaneous symmetry breaking consider a free massless scalar field with a lagrangian (11-34) invariant under the field translations
</>(x) - </>(x)
(11-35)
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