Takaaki Nomura(Saitama univ) Standard(-like) Model from an SO(12) Grand Unified Theory in six-dimensions with S^2 extra space arXiv:0810.0898(Nucl.Phys.B811:109-122,2009) Takaaki Nomura(Saitama univ) collaborator Joe Sato (Saitama univ) Outline Introduction Our approach to GHU The SO(12) model summary 18 July 2009 EPS HEP conference

1. Introduction

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 )

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 )

2.Our approach to GHU

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 :

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

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)

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

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

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

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

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

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

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

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

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

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)

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

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 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

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

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)

3.The SO(12) model

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)

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

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:

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:

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

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!

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!

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

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

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

Appendix Vielbein Killing vectors

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

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

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)

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:

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:

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!

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

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)

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)

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

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

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