Custodial symmetry

Motivation

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

$V_{{SM}}=-\lambda (H^{\dagger }H)+\mu (H^{\dagger }H)^{2}$

responsible for keeping $\rho$ ≈ 1 and insuring small corrections to $\rho$ is called a custodial symmetry.^[1] (Note $\rho$ 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 $\left|H^{\dagger }D_{\mu }H\right|^{2}/\Lambda ^{2}$ 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 $\left|H^{\dagger }D_{\mu }H\right|^{2}/\Lambda ^{2}$ 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 $D^{\mu }H^{\dagger }D_{\mu }H$ or tachyonic mass terms like $H^{\dagger }H$ or self-coupling terms like $\left(H^{\dagger }H\right)^{2}$ (fortunately!) but will prevent $\left|H^{\dagger }D_{\mu }H\right|^{2}/\Lambda ^{2}$.

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.^[2]

See also

• Peskin-Takeuchi parameter
• left-right model
• little Higgs

References

1. P. Sikivie, L. Susskind, M. B. Voloshin and V. I. Zakharov, Nucl. Phys. B 173, 189 (1980).
2. B. Grzadkowski, M. Maniatis, Jose Wudka, "Note on Custodial Symmetry in the Two-Higgs-Doublet Model", arXiv:1011.5228.

External links

• Rodolfo A. Diaz and R. Martínez, "The Custodial Symmetry", arXiv:hep-ph/0302058.