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

2. -parameter in SM (at tree level) (loop level) a few percent
(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.  = V()= V(||2)  is SU(2) doublet： 1 + i 2 3 + i 4  get a vev:
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 Qup - Qdown = +1
equal
Qup
Qdown
Qup
- Qdown = +1 + 0 v
SSB
 =
(Y = +1/2) 0 - v
 =
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. think: if Higgs is not doublet, ...... ?
gauge boson mass (in detail)
= ab ?
= 0 ?
think: if Higgs is not doublet, ?

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

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

13. 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

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

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

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

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

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

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

20. Summary： custodial SU(2) (tree level)  = 1 +  (loop level) vanish
a symmetry for V()
custodial SU(2)
broken by Higgs U(1)Y gauge interaction
Yukawa interactions
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