Lepton Sector in Srednicki

1. Aug 11, 2011

parton

Hi all,

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
$$l, \overline{e}, \varphi$$

in the (SU(2), U(1)) representations
$$(2, -1/2), (1,1), (2, -1/2)$$

Now there is also a Yukawa term of the form
$$\mathcal{L}_{\text{Yuk}} = - y \varepsilon^{ij} \varphi_{i} l_{j} \overline{e} + \text{h.c.}$$

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:
$$\mathcal{L}_{\text{Yuk}} = y \varphi^{j} l_{j} \overline{e} + \text{h.c.}$$

Using $$\varphi^{j} l_{j} = \varepsilon^{ji} \varphi_{i} l_{j} = - \varepsilon^{ij} \varphi_{i} l_{j}$$

we obtain the Yukawa term above with the minus sign.

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

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,
$$(\varphi_{i})^{\dagger} = \varphi^{\dagger} \, ^{i}$$ which would be in the representation
$$(2, +1/2)$$

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