1. Takaaki Nomura(Saitama univ)
Standard(-like) Model from an SO(12) Grand Unified Theory in six-dimensions with S^2 extra space
arXiv: (Nucl.Phys.B811: ,2009)
Takaaki Nomura(Saitama univ)
collaborator
Joe Sato (Saitama univ)
Outline
Introduction
Our approach to GHU
The SO(12) model
summary
18 July EPS HEP conference

2. 1. Introduction

3. Further restriction of a theory is possible
1. Introduction
Gauge-Higgs unification is an attractive scenario describing the Higgs sector
Manton (1979)
Providing origin of the Higgs field
Higgs field is identified as extra-components of gauge field
The Higgs sector is predictive
determined by the gauge interaction in higher-dimensions
Reduced Four-dimensional theory is restricted by
non-trivial boundary conditions of extra-space
Further restriction of a theory is possible
Symmetry condition of gauge field (Manton & Forgacs (1980))
Isometry transformation of extra-space gauge transformation
An extra space should has coset space S/R structure
(ex. SU(2)/U(1) = two sphere )

4. using two-sphere extra-space
1. Introduction
Gauge-Higgs unification is an attractive scenario describing the Higgs sector
Manton (1979)
We construct a Gauge-Higgs unification model with the symmetry condition
using two-sphere extra-space
Providing origin of the Higgs field
Higgs field is identified as extra-components of gauge field
The Higgs sector is predictive
determined by the gauge interaction in higher-dimensions
Reduced Four-dimensional theory is restricted by
non-trivial boundary conditions of extra-space
Further restriction of a theory is possible
Symmetry condition of gauge field (Manton & Forgacs (1980))
Isometry transformation of extra-space gauge transformation
An extra space should has coset space S/R structure
(ex. SU(2)/U(1) = two sphere )

5. 2.Our approach to GHU

6. Gauge theory on 6-dim space-time
2. Our approach to GHU
Theory in 6-dim
Gauge theory on 6-dim space-time
: 4-dim Minkowski space-time
: 2-sphere orbifold
Isometry group
Coordinates:
orbifolding :

7. Fields in the theory Left handed Weyl fermion of SO(1,5) Gauge field
2. Our approach to GHU
Fields in the theory
Left handed Weyl fermion of SO(1,5)
:Left handed Weyl fermion of SO(1,3)
:Right handed Weyl fermion of SO(1,3)
Gauge field
We introduce a background gauge field
: generator
of should be embedded in gauge group G
It is necessary to obtain massless chiral fermion

8. Action of the theory : metric (R:radius) :6-dim gamma matrix
2. Our approach to GHU
Action of the theory
: metric
(R:radius)
:6-dim gamma matrix
:covariant derivative　
Spin connection term (for fermion)

9. Reduction of the theory to 4-dim effective theory
2. Our approach to GHU
Reduction of the theory to 4-dim effective theory
4-dim theory is determined by these conditions
The non-trivial boundary condition of
Restricting gauge symmetry and massless particle contents in four-dimensions
Symmetry condition of gauge field (Manton & Forgacs (1980))
Restricting 4-dim gauge symmetry
and scalar particle contens
The condition to obtain massless fermions
Restricting massless fermion contents in four-dimension

10. The non-trivial boundary condition of
2. Our approach to GHU
The non-trivial boundary condition of
We impose non-trivial boundary condition for
P, P: matrices acting on the representation space of gauge group
Components of P, P is +1 or -1 and

11. Only parity even components have
2. Our approach to GHU
The non-trivial boundary condition of
We impose non-trivial boundary condition for
Only parity even components have
massless mode
P, P: matrices acting on the representation space of gauge group
Components of P, P is +1 or -1 and

12. Symmetry condition of the gauge field
2. Our approach to GHU
Symmetry condition of the gauge field
SU(2) isometry transformation of
Gauge transformation of
should be embedded in G
Infinitesimal form
:Killing vector of
:fields generating gauge transformation

13. Symmetry condition of the gauge field
2. Our approach to GHU
Symmetry condition of the gauge field
SU(2) isometry transformation of
Gauge transformation of
should be embedded in G
Infinitesimal form
:Killing vector of
:fields generating gauge transformation
Isometry transformation on
Gauge transformation

14. Symmetry condition of the gauge field
2. Our approach to GHU
Symmetry condition of the gauge field
SU(2) isometry transformation of
Gauge transformation of
should be embedded in G
Infinitesimal form
:Killing vector of
:fields generating gauge transformation
Isometry transformation on
Gauge transformation
By the condition
Independent of coordinate
Dimensional reduction can be carried out

15. Analysis by mode expansion
2. Our approach to GHU
Analysis by mode expansion
Mode expansion of gauge filed
+: for parity even compontents
- : for parity odd components
Infinitesimal form of SC
Determined by requiring
invariance of BG gauge field
Isometry transformation on
Gauge transformation

16. Ex. for If with i=1 i=2 i=3 Remaining components
2. Our approach to GHU
Ex. for
i=1
i=2
i=3 If
Remaining components
with

17. We can carry our similar analysis for
2. Our approach to GHU
Ex. for
i=1
We can carry our similar analysis for
components
i=2
i=3 If
Remaining components
with

18. Agreement with Manton’s solution for the symmetry condition
2. Our approach to GHU
Remaining components
:scalar fields
Constraints
4-dim gauge group has to commute with in G
charges of the 4-dim scalars are restricted
: generator embedded in G
Agreement with Manton’s solution for the symmetry condition
Manton (1979)

19. Dimensional reduction of the gauge sector
2. Our approach to GHU
Dimensional reduction of the gauge sector
Applying solution of the condition,
Extra-space can be integrated and we obtain…
4-dim gauge-Higgs sector
4-dim Higgs sector with potential term
No massive Kaluza-Klein mode

20. The condition to obtain massless fermions
2. Our approach to GHU
The condition to obtain massless fermions
Positive curvature of
Masses of fermions
in four-dim

21. The condition to obtain massless fermions
2. Our approach to GHU
The condition to obtain massless fermions
Positive curvature of
Masses of fermions
in four-dim
The background
gauge field
cancel

22. The condition to obtain massless fermions
2. Our approach to GHU
The condition to obtain massless fermions
Positive curvature of
Masses of fermions
in four-dim
The background
gauge field
cancel
Spin connection term should be canceled by background gauge field
charges of the 4-dim massless fermion contents are restricted
4-dim fermion sector is obtained by mode expansion
Massive KK modes appear

23. The reduced 4-dim theory
2. Our approach to GHU
The reduced 4-dim theory
Particle contents and gauge symmetry in four-dimensions are determined by
The parity assignment for non-trivial boundary condition
(for gauge field)
The constraints from symmetry condition
The condition for massless fermion
(for fermions)

24. 3.The SO(12) model

25. Set up For gauge group For fileds Gauge group in 6-dim G=SO(12)
3. The SO(12) model
Set up
For gauge group
Gauge group in 6-dim G=SO(12)
U(1) of SU(2)/U(1) is embedded into SO(12) as
For fileds
Two types of left-handed weyl fermion in 32 reps of SO(12)
The parity assignment for P and P in SO(12) spinor basis
(P assignment, P assignment)

26. Gauge symmetry and particl contents in 4-dim
3. The SO(12) model
Gauge symmetry and particl contents in 4-dim
For gauge field and gauge group
Constraints:
( generator )
Gauge group should commute with
For scalar field contents
Constraints:
adj rep of
4-dim scalar contents should be contained in

27. Particle contents of gauge field
3. The SO(12) model
Particle contents of gauge field
Parity assignment for adj SO(12)=66 under
10(2)+10(-2) (66=45(0)+1(0)+10(2)+10(-2))
Do not satisfy constraints:

28. Particle contents of gauge field
3. The SO(12) model
Particle contents of gauge field
Parity assignment for adj SO(12)=66 under
10(2)+10(-2)
Do not satisfy constraints:

29. Particle contents of gauge field
3. The SO(12) model
Particle contents of gauge field
Parity assignment for adj SO(12)=66 under
Symmetry breaking

30. Particle contents of scalar field
3. The SO(12) model
Particle contents of scalar field 66
{(1,2)(3,2,-2), (1,2)(-3,-2,2)}
Only SM Higgs doublet remain!
Particle contents of massless fermion
One generation of massless fermion modes!

31. Analysis of the Higgs potential
3. The SO(12) model
Analysis of the Higgs potential
The Higgs sector in terms of
KE term
potential term
Rewrite in terms of Higgs doublet
(R: radius of the two-sphere)
This potential leads electroweak symmetry breaking!

32. Prediction Vacuum expectation value of Higgs field are obtained
3. The SO(12) model
Prediction
Vacuum expectation value of Higgs field are obtained
in terms of
The radius of the two-sphere
Gauge coupling constant
W boson mass and Higgs mass are also given as

33. Summary We analyzed Gauge Higgs Unification for gauge theory on
with symmetry condition and non-trivial boundary condition
We construct SO(12) model and obtained
One SM generation of massless fermion mode (with RH neutrino)
SM Higgs field and prediction for Higgs mass

34. Future work Extra U(1) symmetry
Utilize to generate mass matrices as family symmetry.
Break by some mechanism
Realistic Yukawa coupling
Realistic mass hierarchy
Realistic fermion mixing
Fermion mass spectrum
Search for a dark matter candidate

35. Appendix
Vielbein
Killing vectors

36. Gauge Higgs Unification(GHU)
1. Introduction
Gauge Higgs Unification(GHU)
Manton (1979)
Gauge sector of high-dim gauge theory
Gauge group G
Higher-dim
Gauge field:
(M=0,1,2,…)
4-dim gauge sector
4-dim Higgs sector
4D-components
Extra-components
Gauge group
H⊂G
4-dim
Gauge field in 4-dim
scalar fields in 4-dim
Interpret as SM Higgs
Gauge group G should be to provide SM Higgs doublet
Model with Simple group G is especially interesting = GUT-like GHU

37. Higgs sector would be predictive
1. Introduction
Gauge Higgs Unification(GHU)
Manton (1979)
Gauge sector of high-dim gauge theory
Gauge group G
Higher-dim
Gauge field:
(M=0,1,2,…)
Higgs sector of four-dimensional theory is determined by gauge interaction of higher-dimensional theory
4-dim gauge sector
4-dim Higgs sector
4D-components
Extra-components
Higgs sector would be predictive
Gauge group
H⊂G
4-dim
Gauge field in 4-dim
scalar fields in 4-dim
Interpret as SM Higgs
Gauge group G should be to provide SM Higgs doublet
Model with Simple group G is especially interesting = GUT-like GHU

38. For massless modes of fermion
3. The SO(12) model
For massless modes of fermion
Constraints:
charge
charge of the massless fermion contents should be…
+1 for left handed fermion
－1 for right handed fermion
For 32 rep of SO(12)
4-dim left-handed contents should be contained in 16(1)
4-dim left-handed contents should be contained in 16(-1)

39. Particle contents in 4-dim: scalar(Higgs) field
3. The SO(12) model
Particle contents in 4-dim: scalar(Higgs) field
Parity assignment for adj SO(12)=66 under
10(2)+10(-2) (66=45(0)+1(0)+10(2)+10(-2))
satisfy constraints:

40. Particle contents in 4-dim: scalar(Higgs) field
3. The SO(12) model
Particle contents in 4-dim: scalar(Higgs) field
Parity assignment for adj SO(12)=66 under
10(2)+10(-2) (66=45(0)+1(0)+10(2)+10(-2))
satisfy constraints:

41. Particle contents in 4-dim: scalar(Higgs) field
3. The SO(12) model
Particle contents in 4-dim: scalar(Higgs) field
Parity assignment for adj SO(12)=66 under
Scalar field in 4-dim
{(1,2)(3,2,-2), (1,2)(-3,-2,2)}
Only SM Higgs doublet remain!

42. Particle contents in 4-dim: fermions
3. The SO(12) model
Particle contents in 4-dim: fermions
Parity assignment for under

43. Particle contents in 4-dim: fermions
3. The SO(12) model
Particle contents in 4-dim: fermions
Parity assignment for under
16(1)
16(-1)
16(1)
16(-1)

44. Particle contents in 4-dim: fermions
3. The SO(12) model
Particle contents in 4-dim: fermions
Parity assignment for under
16(1)
16(-1)
16(1)
16(-1)

45. Particle contents in 4-dim: fermions
3. The SO(12) model
Particle contents in 4-dim: fermions
Parity assignment for under
From the fermion , we obtain

46. Parity assignment for under
3. The SO(12) model
Parity assignment for under
16(1)
16(-1)
16(1)
16(-1)

47. Parity assignment for under
3. The SO(12) model
Parity assignment for under
16(1)
16(1)
16(-1)
16(1)
16(-1)