In particle physics, a symmetry that remains after spontaneous symmetry breaking that can prevent higher-order radiative corrections from spoiling some property of a theory is called a custodial symmetry.
VSM=-\mu(H\daggerH)-λ(H\daggerH)2
\rho
\rho
With one or more electroweak Higgs doublets in the Higgs sector, the effective action term
\left|H\daggerD\muH\right|2/Λ2
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.
Before electroweak symmetry breaking there was a global SU(2)xSU(2) symmetry in the Higgs potential, which is broken to just SU(2) after electroweak symmetry breaking. This remnant symmetry is called custodial symmetry. The total standard model lagrangian would be custodial symmetric if the yukawa couplings are the same, i.e. Yu=Yd and hypercharge coupling is zero. It is very important to see beyond the standard model effect by including new terms which violate custodial symmetry.
The preferred way of preventing the
\left|H\daggerD\muH\right|2/Λ2
H
H
D\muH\daggerD\muH
H\daggerH
\left(H\daggerH\right)2
\left|H\daggerD\muH\right|2/Λ2
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]