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Introduction to Gauge Higgs unification with a graded Lie algebra

Introduction to Gauge Higgs unification with a graded Lie algebra. 2011. 10. 7 @ Academia Sinica , Taiwan Jubin Park (NTHU). Collaboration with Prof. We-Fu Chang. Based on D. B. Fairlie PLB 82,1.

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Introduction to Gauge Higgs unification with a graded Lie algebra

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  1. Introduction to Gauge Higgs unification with a graded Lie algebra 2011. 10. 7 @ Academia Sinica, Taiwan Jubin Park (NTHU) • Collaboration with Prof. We-Fu Chang • Based on D. B. Fairlie PLB 82,1. • G. Bhattacharyya arxiv:0910.5095 [hep-ph] C. Csaki, J. Hubisz and P. Meade hep-ph/0510275

  2. Contents • Brief introduction to a differencebetween 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

  3. 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 Park @ A. Sinica

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

  5. 5D quantum electrodynamics(QED) on S1/Z2 5D Gauge sym. Model setup Boundary conditions (BCs) Periodic BCs Orbifold BCs Jubin Park @ A. Sinica

  6. Kaluza-Klien mode expansion Remnant gauge symmetry 4D shift sym 4D gauge sym. Jubin Park @ A. Sinica

  7. Integrating out fifth dimension Propagators Using a ‘t Hooft gauge. Jubin Park @ A. Sinica

  8. 5D SU(2) example (Non-Abelian case) Lie algebra valued gauge field Boundary conditions (BCs) Projection matrix.c Only diagonal components can have “Zero modes” due to Neumann boundary conditions at two fixed points Gauge sym. Breaking Jubin Park @ A. Sinica

  9. 5D SU(3) example (with 2 scalar doublet) Lie algebra valued gauge field : Gell-Mann martices Boundary conditions (BCs) - Zero modes. Gauge sym. Breaking * Branching Rule Jubin Park @ A. Sinica

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

  11. 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 Park @ A. Sinica

  12. Possible answers for these problems and Goals Jubin Park @ A. Sinica

  13. Possible answers for these problems Wrong weak mixing angle - 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) • Burdman and Y.~Nomura, Nucl. Phys. B656, 3 (2003) : arXiv:hep-ph/0210257]. • R. Coquereaux et.al, CNRSG.~ • I. Antoniadis, K. Benakli and M. Quiros, New J. Phys. 3, 20 (2001) [arXiv:hep-th/0108005]. Abandon the gauge coupling unification scheme . Jubin Park @ A. Sinica

  14. Higgs potential • - 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. • Y. Hosotani, PLB 126, 309, Ann. Phys. 190, 233 • N. Manton, Nucl. Phys. B 158, 141 Jubin Park @ A. Sinica

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

  16. Adding to brane kinetic terms SU(2) U(1) 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 Jubin Park @ A. Sinica

  17. Final 4D effective Lagrangian No mass term of the Higgs because of higher dimensional gauge symmetry Weak mixing angle 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. Jubin Park @ A. Sinica

  18. Higgs potential, Radial modes ~ Massive Nambu-goldstone boson modes ~ Massless (flat direction) Finally, we can get this relation ( with brane Kinetic terms ), We can rewrite the equation with previous relation, Jubin Park @ A. Sinica

  19. 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 Park @ A. Sinica

  20. Phenomenologically viable GHU modelsPhenomenologically viable GHU models Jubin Park @ A. Sinica

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

  22. Model setup : A pure Yang-Mills theory on 6D U(1) SU(2) Covariant derivative and Field strength Jubin Park @ A. Sinica

  23. Covariant derivative of the scalar Hyper charge Effective kinetic term in 4D Kinetic term Potential of scalar Jubin Park @ A. Sinica

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

  25. ? This is not a Lie algebra ( Traceless cond.) • K = -2 case 1. Hyper charge of scalar = +1 2. We can have the same relations in the model, like SM has. Jubin Park @ A. Sinica

  26. ? Supertraceless No zero trace condition because of K=-2, -1-1 + k ≠0 =0 Supertrace tr(a) tr(b) can satisfy usual SU(2) and U(1) Lie algebra commutators can satisfy anticommutators(ACs), and these ACs generatesusual Lie transformation. (Closed) Z2 graded Lie algebra - SU(2|1) • V. G. Kac, Commum. Math. Phys. 53, 31 Jubin Park @ A. Sinica

  27. 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 Park @ A. Sinica

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

  29. 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 Jubin Park @ A. Sinica

  30. 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 Park @ A. Sinica

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