- #1

Aethaeon

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Hello,

Just a few questions about a couple of terms in and not in the SM Lagrangian. I'll talk in particular about these fields, and their representations in SU(3) x SU(2) x U(1)

Q (3,2,1/6) (left-handed quarks, fermion)

U (3,1,2/3) (right-handed up-quarks, fermion)

[tex]\phi[/tex] (1,2,-1/2) (higgs, scalar)

1) Mass terms

There is a term in the Lagrangian of the form:

[tex]Z^a_b \overline{Q_a}\not{\nabla}Q^b [/tex]

Why can't there be a term like

[tex]Z^a_b \overline{Q_a}Q^b [/tex] where a and b are the SU(3) indices

The diagonal of Z would act as mass terms for the Qs

Likewise for the other fermion fields.. I don't see an immediate reason why those terms should be excluded?

2) Generalizations of pre-mass terms

There is a term in the Lagrangian of the form

[tex] Z^a_b \overline{Q_a_{\alpha}} U^b \phi^{\alpha} [/tex]

with a,b SU(3) indices and [itex]\alpha[/itex] the SU(2) index.

The reason this works out is because

[tex] \overline{Q_a_{\alpha}} [/tex]

transforms under [itex](\bar{3},\bar{2},1/6)[/itex], and the two fields together

[tex] U^b \phi^{\alpha} [/tex]

transform under (3,2,-1/6) -- so the interaction conserves all the charges

Why cant there be a more general term that can mix up the SU(2) indices, like

[tex] Z^a_b^{\alpha}_{\beta} \overline{Q_a_{\alpha}} U^b \phi^{\beta} [/tex]

Thanks so much for your time and help..

Just a few questions about a couple of terms in and not in the SM Lagrangian. I'll talk in particular about these fields, and their representations in SU(3) x SU(2) x U(1)

Q (3,2,1/6) (left-handed quarks, fermion)

U (3,1,2/3) (right-handed up-quarks, fermion)

[tex]\phi[/tex] (1,2,-1/2) (higgs, scalar)

1) Mass terms

There is a term in the Lagrangian of the form:

[tex]Z^a_b \overline{Q_a}\not{\nabla}Q^b [/tex]

Why can't there be a term like

[tex]Z^a_b \overline{Q_a}Q^b [/tex] where a and b are the SU(3) indices

The diagonal of Z would act as mass terms for the Qs

Likewise for the other fermion fields.. I don't see an immediate reason why those terms should be excluded?

2) Generalizations of pre-mass terms

There is a term in the Lagrangian of the form

[tex] Z^a_b \overline{Q_a_{\alpha}} U^b \phi^{\alpha} [/tex]

with a,b SU(3) indices and [itex]\alpha[/itex] the SU(2) index.

The reason this works out is because

[tex] \overline{Q_a_{\alpha}} [/tex]

transforms under [itex](\bar{3},\bar{2},1/6)[/itex], and the two fields together

[tex] U^b \phi^{\alpha} [/tex]

transform under (3,2,-1/6) -- so the interaction conserves all the charges

Why cant there be a more general term that can mix up the SU(2) indices, like

[tex] Z^a_b^{\alpha}_{\beta} \overline{Q_a_{\alpha}} U^b \phi^{\beta} [/tex]

Thanks so much for your time and help..

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