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{{Expert-subject|Physics|date=November 2008}}

In [[particle physics]] the extra symmetry of the Higgs potential in the [[Standard Model]]
:<math>V_{SM} = -\lambda (H^\dagger H) + \mu(H^\dagger H)^2</math>
responsible for keeping <math>\rho</math> ≈ 1 and insuring small corrections to <math>\rho</math> is called a '''custodial symmetry'''.<ref>P. Sikivie, L. Susskind, M. B. Voloshin and V. I. Zakharov, Nucl. Phys. B 173, 189 (1980).</ref>
(Note <math>\rho</math> 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 <math>\left|H^\dagger D_\mu H\right|^2/\Lambda^2</math> which generically arises whenever we have new physics [[beyond the Standard Model]] at the scale Λ contributes to the [[Peskin-Takeuchi]] T parameter. However, current precision electroweak measurements restrict Λ to more than a few [[TeV]]. This will not be a problem if we have no new physics right up to at least that scale. However, attempts to solve the [[gauge hierarchy problem]] generically require the addition of new particles below that scale. The preferred way of preventing the nasty <math>\left|H^\dagger D_\mu H\right|^2/\Lambda^2</math> term from being generated is to introduce an [[approximate symmetry]] which  acts upon the Higgs sector. In addition to the gauged SU(2)<sub>W</sub> which acts exactly upon the Higgs doublets, we will also introduce another approximate global SU(2)<sub>R</sub> symmetry which also acts upon the Higgs doublet. The Higgs doublet is now a [[real representation]] (2,2) of SU(2)<sub>L</sub> &times; SU(2)<sub>R</sub> 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 <math>D^\mu H^\dagger D_\mu H</math> or tachyonic mass terms like <math>H^\dagger H</math> or self-coupling terms like <math>\left(H^\dagger H\right)^2</math> (fortunately!) but will outlaw <math>\left|H^\dagger D_\mu H\right|^2/\Lambda^2</math>. On the other hand, such an SU(2)<sub>R</sub> symmetry can never be exact and unbroken because otherwise, the up-type and the down-type Yukawa couplings will be exactly identical. Besides, SU(2)<sub>R</sub> does not map the [[hypercharge]] symmetry U(1)<sub>Y</sub> to itself but this is not too much of a problem because the hypercharge gauge coupling strength is small and in the limit as it goes to zero, we won't have a problem. In the parlance of model building, we say that U(1)<sub>Y</sub> is weakly gauged and this explicitly breaks SU(2)<sub>R</sub>. After the Higgs doublet acquires a nonzero [[vacuum expectation value]], the (approximate) SU(2)<sub>L</sub> &times; SU(2)<sub>R</sub> symmetry is spontaneously broken to the (approximate) [[diagonal subgroup]] SU(2)<sub>V</sub>. This approximate symmetry is called the '''custodial symmetry'''.<ref>B. Grzadkowski, M. Maniatis, Jose Wudka, "Note on Custodial Symmetry in the Two-Higgs-Doublet Model", [http://arxiv.org/abs/1011.5228 arXiv:1011.5228].</ref>

==See also==
*[[Peskin-Takeuchi parameter]]
*[[left-right model]]
*[[little Higgs]]

==References==
{{Reflist}}

[[Category:Electroweak theory]]