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(11-61)
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is a meaningless infrared divergent integral. No subtraction procedure may be devised to circumvent this difficulty without spoiling the fundamental properties of field theory, for instance, positivity of the Hilbert space metric. A massless scalar field theory is undefined in a two-dimensional world due to severe infrared divergences. In the statistical language, fluctuations overcome energy in destroying long-range order in this dimension. A simple argument reveals the nature of this phenomenon. Let us use a discrete classical Heisenberg model on a lattice. Compare the two configurations shown in Fig. 11-7 where the orientation of the "spin" is allowed to vary along the direction of the first axis, say. The action is replaced by the energy proportional respectively to Ea = - I! and to Eb = - I!-l cos ((J/L). The relative weight of these configurations is given by the Boltzmann factor
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Figure 11-7 Two configurations for the classical Heisenberg model on a d-dimensional lattice, One df the directions has been singled out to show the effect of a continuous rotation of the average spin,
For d larger than 2 we see that the (b) configuration has a negligible weight in the thermodynamic limit and for sufficiently low temperature, meaning that order is favored. For d = 2 averaging over fluctuations will destroy this order. Similar arguments may be presented for a discrete symmetry showing that the lower critical dimension is then equal to one.
11-3 CURRENT ALGEBRA
11-3-1 Current Commutators
Weak interactions have led to consider in detail the structure and properties of hadronic currents. Phenomenologically these interactions are well described by an effective current-current lagrangian of the form
se =
JI'(x)Jt(x)
(11-62)
where G is Fermi's constant equal to
G = (1.026
0.001) x
1O- sm;2
(11-63)
The total current J I'(x) is the sum of leptonic (/1') and hadronic (hI') contributions
(11-64)
Disregarding the contribution of the newly discovered massive lepton, the leptonic current involves only negative helicities for the electron e-, muon J1. -, and neutrinos
Ve ,
vI':
(11-65)
The hadronic current itself combines strangeness conserving h~l1s=O) and strangeness changing h~l1s= 1) parts as
SYMMETRIES
= cos 8c h~"'s=O) + sin 8c h~!>'s=l)
::::::
(11-66)
where 8c is the Cabibbo angle equal approximately to
(11-67)
Such a decomposition implies that we can relate the scale of the various components associated to transitions with different quantum numbers. Such a scale will be provided by the nonlinear algebra of current commutators. Each of these currents is a superposition V - A of a vector and an axial part, as was the leptonic current. In the framework of unitary symmetry, they belong to an octet of currents denoted va, A a(a = 1, ... ,8) with
h~"'s=O) = (Vl'l -
iV/) - (A/ - iA/)
h~"'s= 1) = (V/ - iV/) - (A/ - iA/)
(11-68)
The reader should not be confused by the notation AI' used here for an axial current and its meaning as a potential in electrodynamics. Both are traditional. The current-current weak interaction, supplemented by these hypotheses, leads to remarkable universality properties. Consider, for instance, the matrix element measured in neutron f3 decay: (11-69) The form factors Gvand GA on the right-hand side are evaluated at zero momentum transfer due to the smallness of the neutron-proton mass difference. Within 1 percent the observed value of Gv coincides with the corresponding quantity measured in muon decay. The value quoted in (11-63) is the one extracted from muon decay taking radiative corrections into account. The most precise measurements involving strongly interacting particles are usually performed in allowed f3 transitions (0+ -+ 0+) among nuclear states. Care is also taken to correct for various effects, such as radiative corrections, Cabibbo angle, etc. Some recent values are
14 0
Gv/G = 1.006 Gv/G
26AI-+ 26Mg
The fact that Gv(O) is not renormalized by strong interactions finds a natural interpretation if we assume with Gell-Mann and Feynman that the vector current v;. is conserved and in fact generates the hadronic unitary symmetry. In other words, the VI' a (a = 1,2,3) are the components of the isospin current and v;}"'s= 1) is approximately conserved if we neglect SU(3) breaking. The hadronic electromagnetic current is therefore
l I'
= V + vl V.I' 8 -= I'
where for the purpose of comparison we omit the factor e in the definition of fm. This hypothesis is a generalization of the electromagnetic case, where current
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