Understanding the Yukawa Term in Srednicki's Lepton Sector

[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.

Thanks in advance :)

 

[torus]

Not sure if I understand what you mean. The Yukawa coupling constant is just an arbitrary number at this point, it does not matter which sign it has. Its not like the kinetic terms where the prefactor has a specific fixed value. The Yukawa interaction is parameterized by one parameter, you can choose a way to write it. This changes as soon as the Higgs field acquires a vacuum expectation value, because then you can identify the Yukawa coupling y with something like fermion mass over Higgs vev, both well measured quantities. Then you have to care about which sign your mass has, and choose the sign of the Yukawa coupling constant accordingly.

Hope these ramblings help...