Weak lsospin and Hypercharge in Java

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13.1 Weak lsospin and Hypercharge
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Can the weak neutral current (Jp.NC of Section 12.10), taken together with the charged currents (Jp. and Jp.t), form a symmetry group of weak interactions First, we recall the form of the charged currents,
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(13.1)
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13.1 Weak lsospin and Hypercharge
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where the + and - superscripts are to indicate the charge-raising and chargelowering character of the currents, respectively. The subscript L is used to denote left-handed spinors and records the V-A nature of the charged currents. Here, we have used the particle names to denote the Dirac spinors (u. == Ii, U e == e, etc.). We can rewrite these two charged currents in a suggestive two-dimensional form. We introduce the doublet
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and the "step-up" and "step-down" operators 7" =
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(13.3) where the 7" 's are the usual Pauli spin matrices. The charged currents (13.1) then become (with x dependence as in (6.6))
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J/(x) Jp.-(x)
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Anticipating a possible SU(2) structure for the weak currents, we are led to introduce a neutral current of the form
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(13.5) (13.6)
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We have thus constructed an "isospin" triplet of weak currents, with i whose corresponding charges (13.8) generate an S U(2) L algebra
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The subscript L on SU(2) is to remind us that the weak isospin current couples only left-handed fermions. Can the current Jp.3( x) which we have just introduced be identified directly with the weak neutral current of Section 12.10 Unfortunately, we see immediately that this attractive idea does not work; the observed weak neutral current Jp.NC has
Electroweak Interactions
a right-handed component, see (12.97). However, the electromagnetic current is a neutral current with right- as well as left-handed components. For example, for the electron we have from (6.35) (13.10) Note, in passing, that we have omitted the coupling e in our definition ofi;m. This will simplify our discussion of electroweak interactions. Thus, the electromagnetic currentJ~ of (6.5) is written (13.11) where Q is the charge operator, with eigenvalue Q = -1 for the electron. Technically speaking, we say that Q is the generator of a U(l)em symmetry group of electromagnetic interactions (see Section 14.2). We includeJ: m in an attempt to save the SU(2)L symmetry. Note that neither of the neutral currents J/LNC or i;m respects the SU(2) L symmetry. However, the idea is to form two orthogonal combinations which do have definite transformation properties under SU(2)L; one combination, J/L3, is to complete the weak isospin triplet while the second, ilLY' is unchanged by S U(2) L transformations (i.e., is a weak isospin singlet). i/LY is called the weak hypercharge current and is given by
~y,Xt/;,
(13.12)
where the weak hypercharge Y is defined by
r 3 +"2.
(13.13)
That is,
J"em /L
= J3 /L
+ lJ Y 2 /L
(13.14)
Just as Q generates the group U(l)em' so the hypercharge operator Y generates a symmetry group U(lh. Thus, we have incorporated the electromagnetic interaction, and as a result the symmetry group has been enlarged to SU(2)L X U(lh. In a sense we have unified the electromagnetic with the weak interaction. However, rather than a single unified symmetry group, we have two groups each with an independent coupling strength. So, in addition to e, we will need another coupling to fully specify the electroweak interaction. Thus, from an aesthetic viewpoint the unification is perhaps not completely satisfying (see Section 15.7). The proposed weak isospin and weak hypercharge scheme is mathematically an exact copy of the original Gell-Mann-Nishijima scheme for arranging strange particles in S U(2) hadronic isospin multiplets (see Section 2.9). The names" weak isospin" and" weak hypercharge" are taken from this analogy. The SU(2)L X U(lh proposal was first made by Glashow in 1961, long before the discovery of the weak neutral current, and, as we describe in 15, was