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(12.14)
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where g/fi is a dimensionless weak coupling and q is the momentum carried by the weak boson (the factors 1/12 and t are inserted so that we have the conventional definition of g). In contrast to the photon, the weak boson must be massive, otherwise it would have been directly produced in weak decays. Indeed, it turns out that M w - 80 GeY (see 15). In (12.14), we have been cavalier about the spin sum in the boson propagator, see (6.87). However, at the moment, we are interested in situations where q2 Ma, (e.g., /3-decay and p.-decay). Then, (12.14) reverts to (12.11) with
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and the weak currents interact essentially at a point. That is, in the limit (12.15), the propagator between the currents disappears. Equation (12.15) prompts the idea that weak interactions are weak not because g e, but because Ma, is large. If indeed g == e, then at energies O( M w) and above, the weak interaction would become of comparable strength to the electromagnetic interaction. We may think of g == e as a unification of weak and electromagnetic interactions in much the same way as the unification of the electric and magnetic forces in Maxwell's theory of electromagnetism, where
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with eM = e. At low velocities, the magnetic forces are very weak, whereas for high-velocity particles, the electric and magnetic forces play a comparable role.
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The velocity of light c is the scale which governs the relative strength. The analogue for the electroweak force is M w on the energy scale. The unification of the electromagnetic and weak forces is the subject of s 13 and 15.
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Let us use the observed transition rate of the process
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to estimate G. By analo!$y with the QED calculations of Section 6.2, we write the transition amplitude for this process (Fig. 12.2) in the form
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(12.16)
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y 5)1/;p(x)][fv(x)'y"1(1- y5)1/;e(X)] d 4 x,
(12.17)
see (12.10). For this problem, it is easier not to perform the x integration at this stage. Remember that usually we carry out the x integration and obtain (2'17')4 times the "energy-momentum conserving" delta function (see Section 6.2). We then define
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Thus, (12.16) reduces to the form (12.13). In writing down (12.16), we have assumed that the other nucleons in 14 0 are simply spectators to the decaying proton. However, a priori, we cannot ignore the fact the nucleons participating in /3-decay are bound inside nuclei. We also have no reason to believe the idealized form of weak nucleon current, J(Nl, shown in (12.17), since the nucleons themselves are composite objects and not structureless Dirac particles. Despite these problems, it turns out to be quite easy to get an accurate estimate of G. There are several reasons for this. First, the low-energy weak interaction is essentially a point interaction, and we can ignore the longer-range strong interaction effects we just mentioned. Actually, there is a beautiful and more precise justification for this vague argument. The weak current (fp y"1/;,,) and its conjugate (f" y"1/;p)' together with the electromagnetic current (f py"1/; p)' are believed to form an isospin triplet of conserved vector currents. This is referred to as the conserved vector current hypothesis. The intimate connection with the electromagnetic current "protects" the vector part of the weak current from any strong interaction corrections, just as the electromagnetic charge is protected. The axial vector part, fI1Y"y5 yp , will not contribute to the process as we are considering a transition between two JP = 0+ nuclear states,
Nuclear p-Decay
which precludes a change of parity. Moreover, since the process occurs between two J = 0 nuclei, we can safely assume that the nuclear wavefunction is essentially unchanged by the transition. Another simplifying feature is that the energy released in the decay (about 2 MeV) is small compared to the rest energy of the nuclei. We can therefore use nonrelativistic spinors for the nucleons [see (6.11)], and then only yl' with 1L = 0 contributes [see (6.13)]. Thus,
If; =
[u(Pv)y (1- y5)v(Pe)1!t/;t(x)t[;p(x)e-i(p,+p,jOXd 4 x,
(12.18)
where, as remarked after (12.11), the v spinor v( Pe) describes an outgoing positron of momentum Pe' The e+ and p are emitted with an energy of the order of 1 MeV, and so their de Broglie wavelengths are about 10- 11 cm, which is much larger than the nuclear diameter. We can therefore set
and perform the spatial integration of (12.18). Noting the relation between If; and the invariant amplitude 0lL (see Section 6.2), we obtain (12.19) where 2m N arises from the normalization of the nucleon spinors [see (6.13)] and 2/ fi is the hadronic isospin factor for the 14 0 ~ 14N * transition (see Exercise 12.4).
EXERCISE 12.4 Verify the isospin factor fi in (12.19). Note that 14C, 14N*, 14 0 form an isospin triplet, which can be viewed as nn, np, pp, together with an isospin zero 12C core (see Fig. 2.2). Keep in mind that for indistinguishable proton decays, we must add amplitudes, not probabilities.