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Figure 13-15 Lowest-order contributions to the effective nonleptonic lagrangian: (a) gauge boson,
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(b) Higgs meson, (c) tadpole term.
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To order zero in ex and Gp
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ex/Mf", the strong interaction lagrangian will read
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where M is a mass matrix, the origin of which may be related in part or totally to the breaking of G w. Weinberg has shown that suitable redefinitions of the fields always allow to bring M to a diagonal real form without any Ys factor, while keeping the kinetic term invariant. In other words, to order zero in ex, parity and strangeness are naturally conserved while isospin symmetry requires the additional hypothesis ~ = Mu for the d and u quarks. Let us show that parity and strangeness are not violated to order ex, but only to order Gp - ex/Mf",. In order to verify this point we introduce an effective lagrangian for weak nonleptonic interactions computed from lowest-order exchanges (Fig. 13-15):
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2'eff = -gf",
d4 x .I.'!.F'(x,m 2 )<T].(x)] }(O)
+ 2'(b) + 2'(C)
(13-166)
Only the first term corresponding to gauge bosons has been written. The two additional terms represent Higgs-meson exchange 2'(b) and the vacuum expectation value of Higgs bosons leading to a renormalization of the mass matrix M. We expect the contribution of Higgs bosons to be of order ex(m 2 /Mf",) at least, with m a typical hadronic mass. As regards the first term, apart from the massless photon contribution, the remaining part involving the propagation of heavy Wand Z mesons may be analyzed using the short-distance expansion of the product of two currents. The dominant terms of order ex - gf", lead to an additional renormalization of the mass matrix while sub dominant terms contain factors ex/Mf", - G. The matrix M + DM may again be brought to a diagonal real form. In short, to order ex, parity and strangeness are still naturally preserved. This is not the case for isospin invariance. But even to this day attempts at predicting in an absolute way the supposed electromagnetic mass differences have remained without real success. The same short-distance expansion may be used to understand the dynamical selection rules observed in nonleptonic weak transitions, such as the rule!J.! = i, I!J.S I = 1. The dominant contribution arises from dimension-six operators involving four fermion fields. Indeed, operators of dimension three only contribute to a redefinition of the mass matrix and those of dimension four may be absorbed in a wave-function renormalization. Calculations by Gaillard and Lee and by Altarelli and Maiani favour a !J.! = i enhancement with respect to the !J.! = i transitions by a logarithmic factor [In(Mfv/m 2 )J' of the order 5 to 7 according to the models, whereas the observed enhancement factor is rather of order 20. The discussion is made difficult by the lack of absolute normalization of the matrix elements of the operators involved.
13-5-3 Light-Cone Expansion
In order to analyze the corrections to the parton model of deep inelastic phenomena we have to extend the short-distance expansion. The type of generalization required to go over to light-like separations may be anticipated from the free-field case,
QUANTUM FIELD THEORY
as in Eq. (13-112). Omitting the indices carried by the currents, we would like to show the validity of an asymptotic series
J (-2X)J ( -
~2) ~ L C
x2-- O N,1l
,a(X )XIlI '"
XIlP~"'IlN(O)
(13-167)
in terms of operators O~:~"IlN symmetric and traceless in the Lorentz indices. Perturbatively and up to logarithms, we expect the coefficients CN,a(x) to scale as (13-168) with d J the canonical dimension of J. In contradistinction to the previous case, an infinite number of terms contribute to a given behavior near the light cone, in particular to the dominant one. The grouping of terms is made according to what Gross and Treiman have called twist, i.e., the difference (13-169) between the dimension of the operator and its spin. Strictly speaking the latter characterizes the corresponding representation of the homogeneous Lorentz group. The fact that we need such an infinity of terms is welcome since the matrix elements of this product of currents must give a scaling function F(x). The knowledge of such a function is equivalent to an infinite sequence of numbers, its moments for instance. When the structure functions oflepton-hadron collisions are described as absorptive parts of Compton amplitudes the integer N will be an upper bound on the spin in the exchange t channel. The leading contribution arises from operators of lowest twist. For a theory with spin i- fermions, and scalar and gauge bosons this value is two for the diagonal matrix elements of electromagnetic currents. These operators are bilinear in the fields up to covariant derivatives. They read
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