ABSTRACT
The principle ideas of building the Glashow-Weinberg-Salam (GWS) model of the electroweak interaction were stated in Section 11.2 of Advanced Particle Physics Volume 1 (we recommend a reader to look through this section once more). Here we shall consider this theory in detail. Obviously, the first we must do is to argue the choice of a gauge group in the form of SU(2)L × U(1)Y . By the time of the GWS model making, the weak interaction theory has already passed
two stages. At the former, the weak interaction is described by the four-fermion contact Fermi theory. Splendid successes of QED where the interaction is carried by a photon as well as the belief in symmetry of nature suggest that a similar mechanism should lie in the basis of the weak interaction too. These arguments have culminated in an intermediate vector boson (IVB) theory in which two massive charged vector bosonsW± serve as carriers of the weak interaction. Since the IVBs possess an electrical charge, they take part in the electromagnetic interaction. Now, we must have not only a good (renormalizable) weak interaction theory for the IVBs, but a renormalizable electrodynamics as well. However, numerous attempts of building the models with the required properties, remaining beyond the scope of the spontaneous symmetry breaking, were unsuccessful (see, for example, [8, 9] and references therein). So, the correspondence principle demanded that at least three gauge bosons (W± and γ)∗
must exist in a new theory including the weak and electromagnetic interactions. And these bosons have to interact with the weak and electromagnetic currents (JW and Jem). The simplest group with three generators is the SU(2) group. It turns out, however, that the algebra of the JW and Jem currents is not closed with respect to the commutation operation. For simplicity we consider a theory where there is only an electron and a neutrino. Then, the weak and electric charges are defined in the following way:
SW+ (t) = 1
∫ d3xJW0 (x) =
∫ d3xν†e(x)(1 − γ5)e(x), SW− (t) = SW†+ (t), (12.1)
Q(t) =
∫ d3xJem0 (x) = −
∫ d3xe†(x)e(x). (12.2)
Particle and
Using simultaneous commutation relations for fermions
{ψ†i (x, t), ψj(x′, t)} = δijδ(3)(x − x′), we get
[SW+ (t), S W − (t)] = 2S
W 3 (t), (12.3)
where
SW3 (t) = 1
∫ d3x[ν†e(x)(1 − γ5)νe(x) − e†(x)(1 − γ5)e(x)]. (12.4)
Since SW3 6= Q, then SW± and Q do not form a closed algebra. Really, for the operator Q to be a generator of the SU(2) the charges of a complete multiplet must add up to zero, corresponding to the requirement that the generators for SU(2) must be traceless. In the case under consideration, we try to build a doublet from the particles νe and e which do not explicitly satisfy this condition. Besides, the operators SW± have the V −A-form while the operator Q is purely vectorial. To correct the state of affairs we have three possibilities: (i) One may introduce one more gauge boson associated with the operator SW3 in (12.3). Then, these four generators form the SU(2)× U(1) group algebra. (ii) One may enter several gauge bosons. Then, generators connected with these bosons form the algebra of a group that involves the SU(2) group but is wider than in the first case. (iii) One may add new fermions to the multiplet and thus modify the currents in order that the new set of SW± and Q will be closed to form SU(2) under commutation. Experimental data exclude the third possibility. In the GWS model the SU(2) × U(1)
gauge group is used. Now, our next task is to determine quantum numbers of this group. For the sake of simplicity, we are constrained by one fermion generation. Since in gauge interactions the chirality is conserved, we can proceed from independent left-and righthanded fermions. Therefore, the first generation consists of the following 16 two-component spinors:
νeL, eL, νeR, eR, u α L, d
α L, u
α R, d
(in what follows the color index α will be omitted). From (12.1), (12.2), and (12.4) it follows that the weak currents
SW+ =
∫ d3x(ν†eLeL + u
† LdL), S
W† + = S
W − , (12.5)
SW3 = 1
∫ d3x(ν†eLνeL − e†LeL + u†LuL − d†LdL) (12.6)
are the generators of the SU(2) group. From these expressions for the SU(2) generators, it is evident that
lL ≡ ( νeL eL
) , qL ≡
( uL dL
) (12.7)
represent the SU(2)-doublets and νeR, eR, uR, and dR-singlets. Proceed to the U(1) group. This group must be chosen in such a way that the electric
charge
Q =
∫ d3x
( − e†e+ 2
3 u†u− 1
3 d†d
) =
∫ d3x
( − e†LeL − e†ReR +
3 u†LuL+
+ 2
3 u†RuR −
3 d†LdL −
3 d†LdL
) (12.8)
represents a linear combination of the U(1) group generator and the generator SW3 belonging to the SU(2) group. Obviously, the combination
Q− SW3 = ∫ d3x
[ − 1 2
( ν†eLνeL + e
) + 1
( u†LuL + d
) −
−e†ReR + 2
3 u†RuR −
3 d†RdR
] (12.9)
has the property of giving the same quantum number to all members of an SU(2) doublet in (12.7). The direct calculation shows that it commutes with all the SU(2) generators, that is
[Q− SW3 , SWi ] = 0, i = 1, 2, 3. (12.10) This allows us to choose
Y W = 2(Q− SW3 ) (12.11) as the generator of the U(1) group and refer to YW as the weak hypercharge. Unlike the SWi generators, the Y
W generator does not satisfy any nonlinear commutation relations. Its scale, that is, the proportional constant between it and Q− SW3 , may be chosen at will. To get the correct electric charges for particles we need to use Eqs. (12.9) and (12.11) and set
Y W (lL) = −1, Y W (eR) = −2, Y W (νR) = 0
Y W (qL) = 1
3 , Y W (uR) =
3 , YW (dR) = −2
3 .
