Hi all,(adsbygoogle = window.adsbygoogle || []).push({});

I am just reading Srednicki, chapter 88: The Standard Model: Lepton Sector

and I'm not sure if I really understand it.

There are left-handed Weyl fields

[tex] l, \overline{e}, \varphi [/tex]

in the (SU(2), U(1)) representations

[tex] (2, -1/2), (1,1), (2, -1/2) [/tex]

Now there is also a Yukawa term of the form

[tex] \mathcal{L}_{\text{Yuk}} = - y \varepsilon^{ij} \varphi_{i} l_{j} \overline{e} + \text{h.c.} [/tex]

but I don't understand where this minus sign comes from.

I have the following guess: I could also write this term in the form:

[tex] \mathcal{L}_{\text{Yuk}} = y \varphi^{j} l_{j} \overline{e} + \text{h.c.} [/tex]

Using [tex] \varphi^{j} l_{j} = \varepsilon^{ji} \varphi_{i} l_{j} = - \varepsilon^{ij} \varphi_{i} l_{j} [/tex]

we obtain the Yukawa term above with the minus sign.

But if this is really right, [tex] \varphi^{i} [/tex] would be in the [tex] (\overline{2}, -1/2) [/tex] representation, which is equivalent to [tex] (2,-1/2) [/tex]

But is the U(1) quantum number -1/2 uneffected by raising or lowering the index (it is just an SU(2) index, isn't it?) ?

This number would only change, if we consider the Hermitian adjoint,

[tex] (\varphi_{i})^{\dagger} = \varphi^{\dagger} \, ^{i} [/tex] which would be in the representation

[tex] (2, +1/2) [/tex]

I hope someone could tell whether my thoughts are right or wrong.

Thanks in advance :)

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# Lepton Sector in Srednicki

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