Custodial symmetry

Motivation

In particle physics, the extra symmetry of the Higgs potential in the Standard Model

Contents

• Motivation
• Construction
• References

responsible for keeping ρ ≈ 1 and insuring small corrections to ρ is called a custodial symmetry. (Note ρ is a ratio involving the masses of the weak bosons and the Weinberg angle).

With one or more electroweak Higgs doublets in the Higgs sector, the effective action term | H † D μ H | 2 / Λ 2 which generically arises with physics beyond the Standard Model at the scale Λ contributes to the Peskin-Takeuchi T parameter.

Current precision electroweak measurements restrict Λ to more than a few TeV. Attempts to solve the gauge hierarchy problem generically require the addition of new particles below that scale, however.

Construction

The preferred way of preventing the | H † D μ H | 2 / Λ 2 term from being generated is to introduce an approximate symmetry which acts upon the Higgs sector. In addition to the gauged SU(2)_W which acts exactly upon the Higgs doublets, we will also introduce another approximate global SU(2)_R symmetry which also acts upon the Higgs doublet. The Higgs doublet is now a real representation (2,2) of SU(2)_L × SU(2)_R with four real components. Here, we have relabeled W as L following the standard convention. Such a symmetry will not forbid Higgs kinetic terms like D μ H † D μ H or tachyonic mass terms like H † H or self-coupling terms like ( H † H ) 2 (fortunately!) but will prevent | H † D μ H | 2 / Λ 2 .

Such an SU(2)_R symmetry can never be exact and unbroken because otherwise, the up-type and the down-type Yukawa couplings will be exactly identical. SU(2)_R does not map the hypercharge symmetry U(1)_Y to itself but the hypercharge gauge coupling strength is small and in the limit as it goes to zero, we won't have a problem. U(1)_Y is said to be weakly gauged and this explicitly breaks SU(2)_R.

After the Higgs doublet acquires a nonzero vacuum expectation value, the (approximate) SU(2)_L × SU(2)_R symmetry is spontaneously broken to the (approximate) diagonal subgroup SU(2)_V. This approximate symmetry is called the custodial symmetry.