Question

In class we have discussed the Higgs potential V_{o}(\Phi )= -\mu ^{2}\Phi^{\dagger }\Phi+ (\lambda\Phi^{\dagger }\Phi)^{2} where the Higgs field \Phi was assumed to be a complex singlet scalar for simplicity. For the current analysis consider it to be a complex doublet but for again simplicity arguments let us work in the so-called unitary gauge so that \Phi can be represented as \Phi=\binom{0}{(v+h)/\sqrt2} where all the unphysical degrees \Phi Φ are eliminated. Here v stands for the vacuum expectation value of the Higgs field and h for the Higgs boson. a) i) Do the following: i. Minimize the potential V_0 and eliminate \mu^2 in terms of the other parameters (\lambda,v). ii) Find the value of V_0 at the minimum. iii).Identify the mass term th for the Higgs boson h and express it in terms of parameter (\lambda,v). iv). Drive the Feynman rules for the three-point (hhh) and four-point (hhhh) Higgs interactions. Let us extend the above potential V_0 by adding a non-renormalizable term so that V(\Phi)= V_0(\Phi)+ (\Phi^\dagger \Phi)^3/M^2 b) ) Repeat all the subparts given in part (a) for the potential V(\Phi). Show transcribed image text

Answer 1

→ vacum expectation valu, of thang よfiH 2-12 min 20 1/4

万(レth + afus..avs)-워"..was-4,0 2 4

Alus (i) or minimiation of Pefential 6m2 3m2 3m2 4 4

2 ma,8,8 。 2 2 よ 2. 8 4 8