Standard Model and Beyond · Problem 3.35

Gauge Theories

Physics — Particle Physics · mssm-two-higgs-doublets-ssb

Problem

Specialize the example in Section 3.3.8 to the case

\lambda_I = \lambda_{II} = -\lambda_{III}/2 \equiv \lambda > 0

, for which the boundedness condition (3.181) does not hold. This is a simplified version of the two Higgs doublets in the MSSM, in which the

U(1)

is a gauge symmetry. (a) Find the condition on

\mu^2_{I,II}

A

, and

\lambda

for the existence of a minimum of

V

. Assume that

A

is real and non-negative. (b) Show that

A > 0

is a necessary condition to have both

\nu_1 \neq 0

and

\nu_3 \neq 0

. (c) For

A > 0

, what is the condition that the origin (

\nu_{1,3} = 0

) is not a minimum? If that is the case, is the origin a maximum or a saddlepoint? (d) Find explicit expressions for

\nu^2 \equiv \nu_1^2 + \nu_3^2

and

\gamma \equiv \tan^{-1}(\nu_3/\nu_1)

in terms of

\mu^2_{I,II}

A

, and

\lambda

. (e) Calculate the nonzero eigenvalue

\mu^2_A

\mu^2_{Im}

in terms of

A

and

\gamma

. (f) Express the mass-square matrix

\mu^2_{Re}

in terms of

\mu^2_A

\gamma

and

M_Z^2 \equiv 2\lambda\nu^2

(motivated by the MSSM prototype). (g) Prove the sum rule

\mu^2_a + \mu^2_b = \mu^2_A + M_Z^2

, where

\mu^2_{a,b}

are the

\mu^2_{Re}

eigenvalues. (h) Prove that

\mu^2_a \leq M_Z^2 \cos^2 2\gamma

and

\mu^2_b > \mu^2_A + M_Z^2 \sin^2 2\gamma

, where the labelling is such that

\mu^2_a < \mu^2_b

. (i) Show that these inequalities are saturated for large

\mu^2_A/M_Z^2

, and interpret this limit.