Introduction to Gauge Higgs unification with a graded Lie algebra Academia Sinica, Taiwan Jubin Park (NTHU)  Collaboration with Prof. We-Fu Chang  Based on D. B. Fairlie PLB 82,1. G. Bhattacharyya arxiv: [hep-ph] C. Csaki, J. Hubisz and P. Meade hep-ph/

Contents Brief introduction to a difference between the Higgsless and the Gauge Higgs Unification(GHU) model Higgsless VS GHU Simple examples in the Gauge Higgs unification (GHU) on S1/Z2 - 5D QED - 5D SU(2) - 5D SU(3) Well-known problems in the GHU models Possible answers for these problems and Goals Phenomenologically viable GHU models A simplest GHU model with a SU(2|1) symmetry. - Lepton coupling Summary

Alternative models - Higgsless no zero modes SM gauge bosons = First excited modes - Gauge Higgs Unification SM gauge bosons = Zero modes Needs Higgs mechanism in order to break the EWSB. but there is no Higgs potential in 5D. or Hosotani mechanism. too low Higgs mass (or top quark mass) with VEV which is proportional to 1/R Jubin A. Sinica

Simple examples in the Gauge Higgs unification (GHU) Jubin A. Sinica

Jubin A. Sinica 5D quantum electrodynamics(QED) on S1/Z2 Model setup Boundary conditions (BCs)

Jubin A. Sinica Kaluza-Klien mode expansion Remnant gauge symmetry

Jubin A. Sinica Integrating out fifth dimension Using a ‘t Hooft gauge.

Jubin A. Sinica 5D SU(2) example (Non-Abelian case) Lie algebra valued gauge field Boundary conditions (BCs) Only diagonal components can have “Zero modes” due to Neumann boundary conditions at two fixed points

Jubin A. Sinica 5D SU(3) example (with 2 scalar doublet) Lie algebra valued gauge field Boundary conditions (BCs)

Well-known problems in the GHU models Jubin A. Sinica

Well-known problems Wrong weak mixing angle (,, ) No Higgs potential (to trigger the EWSB). - may generate too low Higgs mass (or top quark) even if we use quantum corrections to make its potential. Realistic construction of Yukawa couplings Jubin A. Sinica

Possible answers for these problems and Goals Jubin A. Sinica

Possible answers for these problems - Brane kinetic terms - Violation of Lorentz symmetry ( SO(1,4) -> SO(1,3) ) - Graded Lie algebra (ex. ) - Using a non-simple group. an anomalous additional U(1) (or U(1)s) Jubin A. Sinica  R. Coquereaux et.al, CNRSG.~  Burdman and Y.~Nomura, Nucl. Phys. B656, 3 (2003) : arXiv:hep-ph/ ].  I. Antoniadis, K. Benakli and M. Quiros, New J. Phys. 3, 20 (2001) [arXiv:hep-th/ ].

- Using a non-simply connected extra- dimension ( the fluctuation of the AB type phase – loop quantum correction) - Using a 6D (or more) pure gauge theory. - Using a background field like a monopole in extra dimensional space Jubin A. Sinica  Y. Hosotani, PLB 126, 309, Ann. Phys. 190, 233  N. Manton, Nucl. Phys. B 158, 141

Jubin A. Sinica One solution for wrong weak mixing angle with brane kinetic terms

Adding to brane kinetic terms Jubin A. Sinica We can easily understand that these terms can give a modification to the gauge couplings without any change of given models. From the effective Lagrangian, we can expect this relation Similarly, for the U(1) coupling

Final 4D effective Lagrangian Jubin A. Sinica This number is completely fixed by the analysis of structure constants of given Lie group (or Lie algebra) regardless of volume factor Z if there are no brane kinetic terms in given models.

Finally, we can get this relation ( with brane Kinetic terms ), We can rewrite the equation with previous relation, Jubin A. Sinica

Goals Stability of the electroweak scale (from the quadratic divergences – Gauge hierarchy problem) Higgs potential - to trigger the electroweak symmetry breaking Correct weak mixing Jubin A. Sinica

Phenomenologically viable GHU modelsPhenomenologically viable GHU models Jubin A. Sinica

A simplest GHU model with a SU(2|1) symmetry Jubin A. Sinica

Jubin A. Sinica Model setup : A pure Yang-Mills theory on 6D Covariant derivative and Field strength

Jubin A. Sinica Covariant derivative of the scalar Effective kinetic term in 4D

However, the Higgs mechanism can not happen due to the sign of quadratic term. That is to say, the photon remains massless Jubin A. Sinica 1. Hyper charge of scalar = -3 ° Embedding SU(3) GHU without diagonal components of zero modes of A5 and A6 3. Mixing between diagonal generators 2. A electroweak mixing angle

Jubin A. Sinica This is not a Lie algebra ( Traceless cond.) 1. Hyper charge of scalar = We can have the same relations in the model, like SM has.

Jubin A. Sinica No zero trace condition because of K=-2, k ≠0 Supertraceless can satisfy usual SU(2) and U(1) Lie algebra commutators can satisfy anticommutators(ACs), and these ACs generates usual Lie transformation. (Closed)  V. G. Kac, Commum. Math. Phys. 53, 31

Jubin A. Sinica An general gauge field that couples to the element T of SU(2|1) Infinitesimal transformation under T element of SU(2|1) where

Jubin A. Sinica The field strength F in this model with the SU(2|1) The Kinetic term is The F46, F55, and F66 terms are Note that A is not neither hermitian nor antisymmetric !!!!!!!!

Jubin A. Sinica Finally we can have this interesting(?) potential, Unlike previous Lie gauge, this model can give correct sign of quadratic term to the Higgs potential in order to trigger Higgs mechanism, and also give correct hypercharge +1 to the scalar particle. After the Higgs mechanism, From the VEV, a mass of the Higgs is

Summary The graded Lie algebra in the GHU scheme can give the correct SM-like Lagrangian at low energy. - Correct weak mixing angle. - Needed Higgs potential for Higgs mechanism. - Not too small mass of the Higgs Jubin A. Sinica