Left-handed fermions

Lowest order tests of the SM in the leptonic sector where for each left-handed fermion doublet /l W3 Qf ... 

We consider the general triangle diagram shown in Fig. 9.5, bearing in mind that we must include both the left-handed and right-handed fermion contributions, and also that each fermion labelled i,j,k can either flow towards or away from a particular vertex. (In Fig. 9.5 we show only the left-handed fermions. Similar diagrams occur for the right-handed ones.)... 

Charge conservation limits the number of cases we have to examine. For Z —> W W from (9.3.1) and (9.3.8), the coupling is only to the left-handed fermions and (9.5.18) becomes... 

But = YljQj- Thus we have the remarkable condition that the algebraic sum of the charges of all the left-handed fermions in the theory must be zero ... 

However, all three particles in this equation are fermions with intrinsic spins S = jh. Therefore, we cannot balance the angular momentum in the reaction as written. The spins of the proton and the electron can be coupled to 0 or 1 ft and can also have relative angular momenta with any integral value from the emission process. This simple spin algebra will never yield the half-integral value on the left-hand side of the equation. Another fermion must be present among the products. 

P is a permutation of atoms with the same spin and necessarily must be even because of the restriction [V"(R) is the potential energy] [23]. The exact density matrix will then appear both on the left-hand side of Eq. (5) and implicitly in the restriction on the right-hand side of Eq. (6). This implies that there exists a restriction which does not have a fermion sign difficulty. In the fixed-node approximation, a trial density matrix is used for the restriction on the right-hand side of Eq. (6). 

In general we have gauge vector bosons Wfi coupled gauge-invariantly to the left-handed parts of a set of fermions... 

Although the data (mostly from neutrino experiments) on the c—s coupling are not conclusive, the evidence suggests that it is also left-handed. The apparent connection between quarks and leptons is fascinating. Both sets of doublets are point-like s = 5 fermions their electric charges are quantized in a related way (Q, Q — 1 with Q = 0 for leptons and Q = for quarks) furthermore, the sum of the electric charges of all fermions (in a colour scheme) vanishes as required to cancel triangle anomalies (see Section 9.5.3). It should be noted, however, that serious questions arise if one tries to put the above quark-lepton connection on a quantitative... 

Closer examination of the operators in this equation shows that for Fermion collisions the parameter a must be chosen to be N + l)/2 to make the operator H — a ( PH + HP on the left-hand side Hermitian. 

This can mostly readily be seen by performing the transposition < >, i = "4>r r (i 1) oil the left-hand side, which gives a minus sign. Then the strings ( )k ( )i" and ( )j can be moved to the appropriate places without any further change in sign, since a pair of fermion operators is moved together. 

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