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Understanding Custodial SU(2) Symmetry in Standard Model

Explore the custodial SU(2) symmetry of the Higgs scalar in the Standard Model and its implications. Discover how the custodial symmetry relates to gauge boson masses and Yukawa interactions.

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Understanding Custodial SU(2) Symmetry in Standard Model

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  1. Lecture 5: Custodial SU(2) Symmetry --- custodial SU(2) symmetry of V()

  2. -parameter in SM (at tree level) a consequence of custodial SU(2) symmetry of V() Higgs scalar is SU(2) doublet SU(2)的基础表示 (loop level) a few percent

  3.  is SU(2) doublet： V() has a custodial SU(2)symmetry SU(2)的基础表示 1 + i 2 3 + i 4  = V()= V(||2) V() has a large symmetry: O(4)  SU(2)  SU(2) O(4) symmetry reduce to: O(3)  SU(2)  get a vev: 3 Goldstone bosons  triplet of O(3) V() has a custodial O(3)  SU(2)symmetry

  4.  is SU(2) doublet： 1 + i 2 3 + i 4  = equal Qup Qdown Qup - Qdown = +1 0 v + 0 SSB  = (Y = +1/2) v 0 0 - SSB  = (Y = -1/2) Q cannot SB only 0 can SB

  5.  is SU(2) doublet： 3 Goldstone bosons – triplet of custodial SU(2) (Y = +1/2) Rotate away 3 Goldstone bosons through a SU(2) rotation gauge boson mass

  6. gauge boson mass (Y = +1/2)

  7. Gauge boson mass 是SU(2) 的基础表示

  8. What about several Higgs doublets ? Will MW=MZ cW still hold ? The answer is: yes ! For each Higgs doublet: the derivation process is same ( with ) finally

  9. gauge boson mass (in detail) = ab ? = 0? think: if Higgs is not doublet, ...... ?

  10. Peskin book: 691, 699-700 global symmetry SSB 3 Goldstone bosons with SU(2) custodial symmetry

  11. (1) talk by V. Pleitez (2) hep-ph/0607144 (3) arXiv: 1011.5228 Custodial SU(2) Symmetry

  12. SM has some accidental global symmetries They are not imposed, they are consequence of: • Lorentz invariance • Gauge invariance • Renormalizability • Representation content of the model Examples: Baryon number, Lepton number, approximate chiral symmetries

  13. SSB: Scalar doublet: Potential: bi-doublet: ?

  14. U(1)Y SU(2)L (x) global and local symmetry:

  15. 扩大为 (x) Then L has a large symmetry 的真空态： SSB A1, A2, A3  Z are in a triplet of SU(2)L+R :

  16. Yukawa couplings: Extending custodial symmetry to Yukawa sector Defining bi-doublets: (right-handed neutrinos are needed)

  17. Yukawa interactions take the form: invariant under SU(2)LSU(2)R SSB Then Yukawa interactions have custodial SU(2)symmetry

  18. tree level: loop level: • Fermion loops This correction vanishes in the limit of mt=mb e.g., Higgs loops:

  19. Summary： a symmetry for V() custodial SU(2) broken by Higgs U(1)Y gauge interaction Yukawainteractions custodial SU(2) symmetry for V() (tree level) Higgs scalar is SU(2) doublet  = 1 +  (loop level) turn off U(1)Y （g’=0） (extend custodial SU(2) to gauge interactions) mu = md(extend custodial SU(2) to Yukawa interactions.) vanish

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