An Introduction To Quantum Field Theory - M. Peskin, D. Schroeder (Perseus, 1995) · Problem 20.5

Electroweak Theory: Basics — M. Peskin, D. Schroeder (Perseus, 1995)

Physics — QFT · two-higgs-doublet-model

Problem

20.5 A model with two Higgs fields. (a) Consider a model with two scalar fields ϕ1\phi_1 and ϕ2\phi_2, which transform as SU(2)SU(2) doublets with Y=1/2Y = 1/2. Assume that the two fields acquire parallel vacuum expectation values of the form (20.23) with vacuum expectation values v1v_1, v2v_2. Show that these vacuum expectation values produce the same gauge boson mass matrix that we found in Section 20.2, with the replacement v2→(v12+v22).v^2 \to (v_1^2 + v_2^2). (b) The most general potential function for a model with two Higgs doublets is quite complex. However, if we impose the discrete symmetry ϕ1→−ϕ1\phi_1 \to -\phi_1, ϕ2→ϕ2\phi_2 \to \phi_2, the most general potential is V(ϕ1,ϕ2)=−μ12ϕ1†ϕ1−μ22ϕ2†ϕ2+λ1(ϕ1†ϕ1)2+λ2(ϕ2†ϕ2)2+λ3(ϕ1†ϕ1)(ϕ2†ϕ2)+λ4(ϕ1†ϕ2)(ϕ2†ϕ1)+λ5((ϕ1†ϕ2)2+h.c.).V(\phi_1, \phi_2) = -\mu_1^2 \phi_1^\dagger \phi_1 - \mu_2^2 \phi_2^\dagger \phi_2 + \lambda_1 (\phi_1^\dagger \phi_1)^2 + \lambda_2 (\phi_2^\dagger \phi_2)^2 + \lambda_3 (\phi_1^\dagger \phi_1)(\phi_2^\dagger \phi_2) + \lambda_4 (\phi_1^\dagger \phi_2)(\phi_2^\dagger \phi_1) + \lambda_5 ((\phi_1^\dagger \phi_2)^2 + \text{h.c.}). Find conditions on the parameters μi\mu_i and λi\lambda_i so that the configuration of vacuum expectation values required in part (a) is a locally stable minimum of this potential. (c) In the unitarity gauge, one linear combination of the upper components of ϕ1\phi_1 and ϕ2\phi_2 is eliminated, while the other remains as a physical field. Show that the physical charged Higgs field has the form ϕ+=sin⁡β ϕ1+−cos⁡β ϕ2+,\phi^+ = \sin\beta\, \phi_1^+ - \cos\beta\, \phi_2^+, where β\beta is defined by the relation tan⁡β=v2v1.\tan\beta = \frac{v_2}{v_1}. (d) Assume that the two Higgs fields couple to quarks by the set of fundamental couplings Lm=−λdijQˉLi⋅ϕ1 dRj−λuijQˉLaiϕ2b†uRj+h.c.\mathcal{L}_m = -\lambda_d^{ij} \bar{Q}_L^i \cdot \phi_1\, d_R^j - \lambda_u^{ij} \bar{Q}_{La}^i \phi_{2b}^\dagger u_R^j + \text{h.c.} Find the couplings of the physical charged Higgs boson of part (c) to the mass eigenstates of quarks. These couplings depend only on the values of the quark masses and tan⁡β\tan\beta and on the elements of the CKM matrix.

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