QUANTUM FIELD THEORY in .NET

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528 QUANTUM FIELD THEORY
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conservation implies a universal charge renormalization. We recall from Chaps. 7 and 8 that the Ward identity expressing the conservation law implied that the charge renormalization was entirely due to vacuum polarization. All fermion interactions resulted only in the wave function renormalization constant Z2 = Zl'
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Therefore some components of the vector hadronic weak current and the isovector part of the electromagnetic current are members of the same multiplet and simultaneously conserved. This conserved vector current (evC) hypothesis of Gell-Mann and Feynman may be checked by comparing the weak and electromagnetic decay rates of members of the same isospin multiplet. In particle physics we find a prediction for the n+ -+ nO + e+ + v and n- -+ nO + e- + v transitions. Due to the smallness of the momentum transfer, the amplitudes are directly normalized, through an isospin rotation, by the corresponding matrix elements of the electric charge. This leads to the rates
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r 1t -+1t0+\ev) =
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where terms of order m;/(m n+ experimental value
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= 0.45 S-l
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mno)2 ~ 10- 2 have been neglected. This is to be compared to the [' n ~no+(ev) = (0.39
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Accurate tests can also be performed in nuclear physics.
In contradistinction, the axial current does not appear to be conserved, even in the limit of an exact SU(3) in variance. Consequently, the axial coupling constant as measured in f3 decay is not equal to the vector one:
GA G = 1.22
0.02
(11-70)
It is, however, conceivable to study a limit where axial currents would be conserved,
at least the strangeness conserving ones. This would correspond to an additional chiral SU(2) x SU(2) in variance group, generated by the charges Qa and Q5 a (a = 1,2,3). If the fundamental dynamical variables suitable for a description of hadrons include the quark fields, an explicit expression may be derived for the currents
A/(x)
= q(X)YI'Y5
~a q(x)
(11-71)
We may then obtain equal-time commutation relations for the currents as well as for the corresponding charges:
Qa(t) Q5 a(t)
d 3 x Voa(x, t)
(11-72)
d x Aoa(x, t)
Gell-Mann has postulated that these commutation relations derived from the quark model remain valid independently of this assumption on the hadronic substructure. If SU(3) is not exactly valid some of the charges may depend on
SYMMETRIES
time, while the equal-time algebra remains time independent:
[Qa, Qb] [Qa, Qsb] [Qsa, Qsb]
ifabcQc ifabcQsc ifabcQc
(11-73)
We recognize that the commutations rules are those of the Lie algebra of the group SU(3) x SU(3). This is readily verified by constructing the left- and righthanded combinations (11-74) which fulfil
[Q a, Q b] = ifabcQ c
[Q+ a, Q_ b]
(11-75)
The ordinary unitary group is the diagonal subgroup of SU(3) x SU(3), and parity exchanges the two sets of charges (11-76) The charge algebra gives its meaning to the universality concept since it is basically nonlinear. For instance, from (11-77) it follows that the matrix element ofthe left-hand side entering a weak amplitude is universally normalized with reference to the isospin. The relations (11-73) may be generalized in two steps. First we may write commutation relations between charges and currents expressing the behavior of V/ and A/ under the SU(3) x SU(3) transformations. This abstracts from the quark model the fact that currents belong to a representation (1, 8) EB (8, 1):
[Qa(t), V/(x, t)]
ifabc V/(x, t)
[Qa(t), Al'b(X, t)] = ifabcA/(X, t) [Qsa(t), V/(x, t)] = - ifabcA/(X, t) [Qsa(t), A/(x, t)] = ifabc V/(x, t)
(11-78)
From Eqs. (11-78) an integral over space allows us to recover the charge commutation relations. In a second step the quark model suggests writing equal-time commutators for the time components:
[Voa(x, t), VOb(y, t)] = ifabc VOC(x, t)!5 3 (x - y) [Yo a(x, t), Aob(y, t)] = ifabcAOC(X, t)!5 3(x - y) [Aoa(x, t), AOb(y, t)]
(11-79)
ifabc VOC(x, t)!5 (x - y)
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