The term for Yukawa coupling between the Higgs and down quarks is given below. $$ \mathcal{L} = -(y_{e}\bar{d_{R}}\Phi^{\dagger}Q_{L}) + h.c. $$ My question is why does it take this form? Specifically, why is the right handed down quark left as a singlet but the left handed down quark is given as a doublet with the left handed up quark. Why isn't the right handed down quark given as a doublet with the right handed up quark or why don't we have both the left and right handed down quarks given as singlets? As an aside, why does the equivalent term for the up quark use the conjugate Higgs?
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As you learn about the SM, all your questions should have self-evident answers. Any good text on the SM would prevent them from being asked. Anyway...
why is the right handed down quark left as a singlet but the left handed down quark is given as a doublet with the left handed up quark. Why isn't the right handed down quark given as a doublet with the right handed up quark or why don't we have both the left and right handed down quarks given as singlets?
The term you wrote generates only the mass term for the d quark. The term for the u quark mass is in your "aside" following question. Primarily, this is an "afterthought" of the gauge theory of the weak interactions. Both such Yukawa terms allow for the respective gauge invariance baked into the Standard Model. Why this type of gauge invariance?
From the Feynman—Gell-Mann V-A tabulation of the weak charged currents observed experimentally, the starting point of such a gauge theory is the weak isospin connection of the left-chiral u,d quarks. They are the ones entering the weak doublet and interconverting by the charged weak current, an experimental observation.
There is no compelling reason to include the right-handed quarks in the gauge theory, except we know that (alas!?) quarks have a mass, so the left-chiral components must connect to the right components. The Yukawa term you wrote down achieves that, virtually uniquely, and connects the left-d to the right-d through the v.e.v. of the Higgs field in the lower component. Its major achievement is that it is invariant under left-SU(2) and also weak hypercharge (a cockeyed-chirality charge, as you should have learned in your course).
As an aside, why does the equivalent term for the up quark use the conjugate Higgs?
It is again inevitable, if you are to give mass to the up quark, as well. You appreciate that the previous Yukawa term, the one you wrote, has the constant v.e.v. for the $\Phi^\dagger$ in the lower component, so it attaches to the lower entry of the left doublet, the d. For the u mass term, by contrast, you need the constant term to attach to the upper component of the doublet, instead, which the conjugate term $\tilde \Phi^\dagger$ achieves.
Both such Yukawa terms are invariant under weak isospin (SU(2)) and hypercharge (U(1)), very tightly, virtually miraculously; beginning students marvel at the devilish fit, by the hours.
Cosmas Zachos
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