The Higgs boson as a five-dimensional gauge field Francesco Knechtli with Maurizio Alberti, Nikos Irges and Graham Moir Wuppertal – Athens – Cambridge DESY Theory Workshop, Hamburg, 28 Sep 2016 Motivation Orbifold Model Phase diagram Spectrum Dimensional Reduction LCPs Conclusions Motivation Gauge-Higgs Unification I Origin of the potential responsible for the Brout-Englert-Higgs mechanism is unknown. I In Gauge-Higgs Unification (GHU) models the Higgs field is associated with some extra-dimensional components of a gauge field [ Manton, 1979 ]. I In the case of one extra dimension the Higgs potential is zero at tree level and is generated only through quantum effects. I At 1-loop, Higgs mass is finite. Fermions are necessary to obtain a symmetry breaking minimum through the Hosotani mechanism [ Hosotani, 1983 ]. I What happens non-perturbatively? lattice study. −! F. Knechtli, Higgs boson from 5d 1/14 Motivation Orbifold Model Phase diagram Spectrum Dimensional Reduction LCPs Conclusions Lattice Formulation Wilson Gauge action for bulk gauge group SU(2): orb β4 X β5 X S = w tr 1 P4 + tr 1 P5 W 2 · f − g 2 f − g P4 P5 Boundary links satisfy gUg−1 = U with g = iσ3 − ( 1 plaquette on boundary w = 2 1 otherwise 5-d Orbifold U(1) U(1) I bare anisotropy is t p k=1,2,3 γ = β5/β4 a4=a5 5 ' I N5 is number of links n5 = 0 n5 = N5 in the fifth dimension SU(2) I N5a5 = πR, R radius of extra dimension. F. Knechtli, Higgs boson from 5d 2/14 Motivation Orbifold Model Phase diagram Spectrum Dimensional Reduction LCPs Conclusions Lattice operators: Higgs l g−1 g l + x = 0 x = π R 55 5 Scalar Polyakov loop (defined at n = (nµ; 0)) p(n) = l(n) g ly(n) gy Higgs field y 1 1 2 2 h(n) = [p(n) p (n); g]=(4N5) A σ + A σ − ∼ 5 5 Higgs operators X y X (n0) = tr [hh ] ; (n0) = tr [p] H P n1;n2;n3 n1;n2;n3 F. Knechtli, Higgs boson from 5d 3/14 Motivation Orbifold Model Phase diagram Spectrum Dimensional Reduction LCPs Conclusions Lattice operators: Gauge boson t α k=1,2,3 g 5 α π x 55 = 0 x 5 = R Gauge boson operators (defined at n = (nµ; 0)) X ^ y (n0; k) = tr [g U(n; k) α(n + a4k) U (n; k) α(n)] Z n1;n2;n3 0 X ^ y 0 y (n0; k) = tr [g U(n; k) l(n + a4k) U (n ; k) l (n)] Z n1;n2;n3 0 α is the SU(2) projection of h; n = (nµ;N5) F. Knechtli, Higgs boson from 5d 4/14 Motivation Orbifold Model Phase diagram Spectrum Dimensional Reduction LCPs Conclusions Reminder Requirements for a viable model it should reproduce the right physics. the fifth dimension should be “hidden”. the physics should be cut-off independent. F. Knechtli, Higgs boson from 5d 5/14 Motivation Orbifold Model Phase diagram Spectrum Dimensional Reduction LCPs Conclusions Phase Diagram Notes 2 Bulk-driven phase transition On the torus Boundary-driven 1.8 1.8 phase transition Measured point I confined and 1.6 Coulomb phase. 1.4 1.4 On the orbifold 1.2 Higgs Phase 5 I One more phase: 1 β 1.0 U(1) gauge links 0.8 deconfine 0.6 0.6 Confined Phase separately. 0.4 Hybrid Phase I No compactification observed at γ > 1. 0.2 0.2 0 I Interesting physics 1.5 2 2.5 3 1.5 2.0 2.5 3.0 is found at γ < 1. β4 First order phase transitions [ Alberti, Irges, FK and Moir, 1506.06035 ] F. Knechtli, Higgs boson from 5d 6/14 1.4 β4=2.1 L = 32 1.2 L = 48 I But, in proximity of the ρ 1.0 0.8 Higgs-Hybrid phase β4=2.7 1.6 0.6 transition: ρ > 1! 1.6 L = 32 1.41.4 L = 48 0.4 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 bulk-driven phase transition 1.2 2.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.4 1.2 1.4 boundary-driven phase transition β5 1 1.3 ρ 1.0 ρ < 1 β =3.0 1 < ρ < 1.3 0.8 4 1.21.2 0.8 L = 32 ρ > 1.3 L = 32 1.6 0.6 1.1 5 simulated point 0.6 L = 48 1.4 L = 48 β 0.4 1.01 0.5 1.011.2 1.51.5 2.02 2.52.5 3.03 0.9 β5 ρ 1.0 0.80.8 0.8 0.7 1.8 2 2.2 2.4 2.6 2.8 3 0.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 0.5 3.5 1 1.5 2 2.5 3 3.5 β4 3.0 0.53.5 1.0 1.5 2.0 2.5 3.0 β5 [ Alberti, Irges, FK and Moir, 1506.06035 ] Motivation Orbifold Model Phase diagram Spectrum Dimensional Reduction LCPs Conclusions Spectrum 1.41.4 β4=1.9 1.3 Mass ratios L = 32 1.21.2 1.1 L = 48 I In general, in the Higgs 1.01 ρ 0.9 2 phase we measure 0.80.8 0.7 Bulk-driven 1.8 mH 0.60.6 phase transition 1.8 ρ = < 1. 0.5 mZ 0.4 Boundary-driven 1.6 0.4 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 Higgs Phase1.2 1.3 1.4 1.5 1.6 1.7phase1.8 transition1.9 2.0 Measured 1.4 β5 1.4 point 1.2 5 1 β 1.0 0.8 0.60.6 Confined Phase 0.4 Hybrid Phase 0.20.2 0 1.51.5 2.02 2.52.5 3.03 β4 F. Knechtli, Higgs boson from 5d 7/14 1.4 β4=2.1 L = 32 1.2 L = 48 ρ 1.0 0.8 β4=2.7 1.61.6 0.6 L = 32 1.41.4 L = 48 0.4 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 bulk-driven phase transition 1.2 2.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.4 1.2 1.4 boundary-driven phase transition β5 1 1.3 ρ 1.0 ρ < 1 β =3.0 1 < ρ < 1.3 0.8 4 1.21.2 0.8 L = 32 ρ > 1.3 L = 32 1.6 0.6 1.1 5 simulated point 0.6 L = 48 1.4 L = 48 β 0.4 1.01 0.5 1.011.2 1.51.5 2.02 2.52.5 3.03 0.9 β5 ρ 1.0 0.80.8 0.8 0.7 1.8 2 2.2 2.4 2.6 2.8 3 0.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 0.5 3.5 1 1.5 2 2.5 3 3.5 β4 3.0 0.53.5 1.0 1.5 2.0 2.5 3.0 β5 [ Alberti, Irges, FK and Moir, 1506.06035 ] Motivation Orbifold Model Phase diagram Spectrum Dimensional Reduction LCPs Conclusions Spectrum 1.41.4 β4=1.9 1.3 Mass ratios L = 32 1.21.2 1.1 L = 48 I In general, in the Higgs 1.01 ρ 0.9 2 phase we measure 0.80.8 0.7 Bulk-driven 1.8 mH 0.60.6 phase transition 1.8 ρ = < 1. 0.5 mZ 0.4 Boundary-driven 1.6 0.4 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 Higgs Phase1.2 1.3 1.4 1.5 1.6 1.7phase1.8 transition1.9 2.0 I But, in proximity of the Measured 1.4 β5 1.4 Higgs-Hybrid phase point 1.2 5 transition: ρ > 1! 1 β 1.0 0.8 0.60.6 Confined Phase 0.4 Hybrid Phase 0.20.2 0 1.51.5 2.02 2.52.5 3.03 β4 F. Knechtli, Higgs boson from 5d 7/14 1.4 β4=2.1 L = 32 1.2 L = 48 ρ 1.0 0.8 β4=2.7 1.61.6 0.6 L = 32 1.41.4 L = 48 0.4 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 bulk-driven phase transition 1.2 2.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.4 1.2 1.4 boundary-driven phase transition β5 1 1.3 ρ 1.0 ρ < 1 β =3.0 1 < ρ < 1.3 0.8 4 1.21.2 0.8 L = 32 ρ > 1.3 L = 32 1.6 0.6 1.1 5 simulated point 0.6 L = 48 1.4 L = 48 β 0.4 1.01 0.5 1.011.2 1.51.5 2.02 2.52.5 3.03 0.9 β5 ρ 1.0 0.80.8 0.8 0.7 1.8 2 2.2 2.4 2.6 2.8 3 0.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 0.5 3.5 1 1.5 2 2.5 3 3.5 β4 3.0 0.53.5 1.0 1.5 2.0 2.5 3.0 β5 [ Alberti, Irges, FK and Moir, 1506.06035 ] Motivation Orbifold Model Phase diagram Spectrum Dimensional Reduction LCPs Conclusions Spectrum 1.41.4 β4=1.9 1.3 Mass ratios L = 32 1.21.2 1.1 L = 48 I In general, in the Higgs 1.01 ρ 0.9 2 phase we measure 0.80.8 0.7 Bulk-driven 1.8 mH 0.60.6 phase transition 1.8 ρ = < 1.