for each pion propagator with four-momentum k and isospin indices i,j.

(Fix k by momentum conservation.)

Rule 3: for fermions, assemble the incoming fermion spinors, vertex operators, and outgoing fermion spinors in order along each fermion line to make a well-formed matrix element. In particular:

• multiply from the left by u(p, s) for each outgoing fermion with momentum p and spin s.

• multiply from the right by u(p, s) for each incoming fermion with momentum p and spin s.

Rule 4: antisymmetrize between identical fermions in the final state.

Except for Rule 3 we have encountered versions of all of these rules before. Note that Rule 1 assumes a different form because of the spin-isospin dependence of the ttNN interaction, and Rule 2 includes the isospin of the exchanged pion. Rule 3 is new; in cases where the initial or final state particles have spin or other quantum numbers, there will always be a spinor or a vector carrying this information which will be a part of the M-m&tnx.

Remarks and Phenomenology

(i) The angular dependent part is referred to as the tensor force. Its structure,

clearly displays the L = 2 character of the tensor force and shows that it averages to zero when integrated over solid angles,

Note also that the tensor potential is highly singular at short distances (~ p-).

(ii) The properties of the central potential can be inferred from the values of eri <r2 and ri ■ r2. If the total spin of the state is S and the total isospin is I, then i-<x2=2[S(S+l)-§] = | 1 { 1 '

Hence the central part of the potential is attractive and has the same strength in the two ¿"-states, but is repulsive in the P-states. Using the spectroscopic notation introduced in Sec. 8.6,

s = |
0, |
1 = |
1 |
(<7 |

s = |
1, |
I = |
0 |
(* |

s = |
0, |
J = |
0 |
(<r, |

s = |
1, |
I = |
1 |
(TI |

attractive

(iii) The central force obtained from the OPE is about 10 times smaller than the central force which is inferred from a phenomenological analysis of NN scattering data. Furthermore, the empirical central force is stronger (more attractive) in 'So than it is in 3Si, yet the only bound state, the deuteron, is a mixture of 3Si - :iD1. The reason that the 3 Si — 3 D\ channel is more tightly bound than the 'So channel is due to the tensor force, which is attractive in the 3 Si -3Di channel (but zero in the 'So channel) and provides the necessary binding. It turns out that the tensor force is well described by the OPE potential.

The study of the nuclear force continues to be a problem of current research. The approach discussed in this section can be extended by adding the exchange of other mesons; such a model is referred to as a one-boson exchange (OBE) model. It has been found that the force can be very well parameterized by an OBE model with pion, scalar, and vector meson exchanges.

9.10 ELECTROWEAK DECAYS

9.10 ELECTROWEAK DECAYS

For a final example, we consider the decay of the charged pion. This decay takes place through the weak charged current, which couples the charged pion with leptons. First, we will discuss these charged currents, and then we will compute the decay of the 7r+.

The weak interactions (which are unified with the electromagnetic interactions, as discussed in Sec. 15.4) are now known to include both charged and neutral currents, but in this section we will discuss the charged currents only. These are interactions of the generic 4>3 structure, and add the following interaction term to the Lagrangian:

where gelt is the effective weak coupling constant,* W^ is a complex (positively charged) vector field, and J™ is the weak charged current. The field W^ will be described by the free Lagrangian

Note that this Lagrangian density is almost identical to the one introduced in Sec. 2.5, except that it is constructed from complex vector fields and, as in the charged <p3 theory, must therefore be regarded as describing two real fields. Hence the Lagrangian density (9.127) is twice as large as the neutral density (2.39) (see Prob. 7.3). Other aspects of the massive spin one theory are the same; in particular, the Wfj, field satisfies the Lorentz condition and the wave equation with mass term d^W" = 0

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