The Higgs Boson As a Five-Dimensional Gauge Field

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 · { − } 2 { − } 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 l†(n) g† Higgs ﬁeld † 1 1 2 2 h(n) = [p(n) p (n), g]/(4N5) A σ + A σ − ∼ 5 5 Higgs operators

X † 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 ˆ † (n0, k) = tr [g U(n, k) α(n + a4k) U (n, k) α(n)] Z n1,n2,n3 0 X ˆ † 0 † (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 phase1.7 1.8 transition1.9 2.0 Measured 1.4 β5 1.4 point

1.2 5

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 β4=2.7 1.61.6 L = 32

1.41.4 L = 48 bulk-driven phase transition 1.2 1.4 1.2 1.4 boundary-driven phase transition

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 1.4 β4Bulk-driven=2.1 1.8 mH 0.60.6 phase transition L = 32 1.8 ρ = < 1. 0.5 mZ 1.2 0.4 Boundary-driven 1.6 L = 48 0.4 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 Higgs Phase1.2 1.01.3 1.4 1.5 1.6 phase1.7 1.8 transition1.9 2.0 I But, in proximity of the ρ Measured 1.4 β5 1.4 Higgs-Hybrid phase 0.8 point 1.2 0.6 5 transition: ρ > 1! 1 0.4 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 β 1.0 2.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 0.8 β5 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 bulk-driven phase transition 1.4 1.4 boundary-driven phase transition 1.3 ρ < 1 1 < ρ < 1.3 β4=3.0 1.21.2 L = 32 ρ > 1.3 L = 32 1.6

1.1 5 simulated point L = 48 1.4 L = 48 β 1.01 1.2 0.9 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 1.4 β4Bulk-driven=2.1 1.8 mH 0.60.6 phase transition L = 32 1.8 ρ = < 1. 0.5 mZ 1.2 0.4 Boundary-driven 1.6 L = 48 0.4 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 Higgs Phase1.2 1.01.3 1.4 1.5 1.6 phase1.7 1.8 transition1.9 2.0 I But, in proximity of the ρ Measured 1.4 β5 1.4 0.8 point Higgs-Hybrid phase β4=2.7 1.2 1.6 0.6 5 transition: ρ > 1! 1.6 L = 32 1.4 1 1.4 L = 48 0.4 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 β 1.0 1.21.2 2.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 0.8 β5 ρ 1.01

0.6 Conﬁned Phase 0.8 0.6 0.8 0.60.6 0.4

0.4 Hybrid1 Phase1.5 2 2.5 3 0.5 1.0 1.5 2.0 2.5 3.0 0.20.2 β5

0 1.51.5 2.02 2.52.5 3.03 β4

F. Knechtli, Higgs boson from 5d 7/14 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 1.4 β4Bulk-driven=2.1 1.8 mH 0.60.6 phase transition L = 32 1.8 ρ = < 1. 0.5 mZ 1.2 0.4 Boundary-driven 1.6 L = 48 0.4 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 Higgs Phase1.2 1.01.3 1.4 1.5 1.6 phase1.7 1.8 transition1.9 2.0 I But, in proximity of the ρ Measured 1.4 β5 1.4 0.8 point Higgs-Hybrid phase β4=2.7 1.2 1.6 0.6 5 transition: ρ > 1! 1.6 L = 32 1.4 1 1.4 L = 48 0.4 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7

β bulk-driven 1.0 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 0.8 phase transition β5 1 1.3 ρ 1.0 ρ < 1 β =3.0 1 < ρ < 1.3 0.8 4 1.21.2 0.6 Conﬁned Phase 0.8 L = 32 0.6 ρ > 1.3 L = 32 1.6 0.6 1.1

5 simulated point 0.6 L = 48 0.4 L = 48 1.4 β

0.4 1 Hybrid1 Phase1.5 2 2.5 3 1.0 0.5 1.01.2 1.5 2.0 2.5 3.0 0.2 0.9 β5 0.2 ρ 1.0 0.80.8 0 0.8 1.5 2 2.5 3

0.7 1.8 2 2.2 2.4 2.6 2.8 3 0.6 1.8 1.5 2.0 2.2 2.4 2.6 2.02.8 3.0 2.5 3.0 0.5 3.5 1 1.5 2 2.5 3 3.5 β4 β 3.0 0.53.5 1.0 1.5 2.0 2.5 3.0 4 β5 [ Alberti, Irges, FK and Moir, 1506.06035 ]

F. Knechtli, Higgs boson from 5d 7/14 Motivation Orbifold Model Phase diagram Spectrum Dimensional Reduction LCPs Conclusions Spontaneous symmetry breaking

Elitzur’s theorem [ Elitzur, 1975 ] In the gauge-invariant lattice theory, spontaneous symmetry breaking (SSB) must originate from the breaking of a global symmetry. Stick symmetry

Global transformation [ Ishiyama, Murata, So and Takenaga, 0911.4555 ] ( U(n = 0, 5) g−1 U(n = 0, 5) 5 s 5 g = iσ1 → −1 with s U(n5 = 0, µ) g U(n5 = 0, µ) gs − → s Gauge boson operators , 0 are odd under the stick sym- Z Z metry, they are the order parameters of SSB [ Irges and FK, arXiv:1312.3142 ]

F. Knechtli, Higgs boson from 5d 8/14 Motivation Orbifold Model Phase diagram Spectrum Dimensional Reduction LCPs Conclusions Gauge-Higgs Uniﬁcation

Gauge-Higgs Uniﬁcation on the Orbifold Does it work?

Requirements of 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 9/14 I But near the PT the boundary remains

four-dimensional. 0.30.3

0.298 I The ﬁtted Yukawa 0.2960.296 masses agree with 0.294

0.292 0.292V(r) Monte Carlo data the measured one. V(r) 0.29 4 4-d Yukawa a 0.2880.288 4-d CoulombMC data 4d Yukawa fit 0.286 5d Yukawa fit dimensional reduction 5-d Coulomb4d Coulomb fit 5d Coulomb fit ⇒ 0.2840.284 5-d Yukawa

via localization 0.282 3 4 5 6 7 8 9 10 11 3 4 5 6 7r 8 9 10 11 r/a4 [ Alberti, Irges, FK and Moir, 1506.06035 ]

Motivation Orbifold Model Phase diagram Spectrum Dimensional Reduction LCPs Conclusions Dimensional Reduction

Higgs phase 0.640.64 0.635

0.63 I The orbifold’s SU(2) 0.63 0.625

0.62 bulk feels the ﬁfth 0.62V(r) Bulk-driven 0.615 1.8 V(r) Monte Carlo data

4 phase transition

0.61 4-d Yukawa 1.8dimension. a 0.61 Boundary-driven 1.6 0.605 4-d Coulomb phase transitionMC data 4d Yukawa fit Higgs Phase 5-d Coulomb 5d Yukawa fit 0.6 4d Coulomb fit Measured 5d Coulomb fit 1.4 0.60 5-d Yukawa 1.4 0.595 point 3 4 5 6 7 8 9 10 11 r 1.2 3 4 5 6 7 8 9 10 11 5

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 10/14 Motivation Orbifold Model Phase diagram Spectrum Dimensional Reduction LCPs Conclusions Dimensional Reduction

Higgs phase 0.640.64 0.635

0.63 I The orbifold’s SU(2) 0.63 0.625

0.62 bulk feels the ﬁfth 0.62V(r) Bulk-driven 0.615 1.8 V(r) Monte Carlo data

4 phase transition

0.61 4-d Yukawa 1.8dimension. a 0.61 Boundary-driven 1.6 0.605 4-d Coulomb phase transitionMC data 4d Yukawa fit Higgs Phase 5-d Coulomb 5d Yukawa fit 0.6 4d Coulomb fit I But near the PT the Measured 5d Coulomb fit 1.4 0.60 5-d Yukawa 1.4 0.595 point 3 4 5 6 7 8 9 10 11 boundary remains r 1.2 3 4 5 6 7 8 9 10 11 5

four-dimensional. 0.3 1

β 0.3

1.0 0.298 I The0.8 ﬁtted Yukawa 0.2960.296

0.294 masses0.6 Conﬁned agree Phase with 0.6 0.292 0.292V(r) 0.4 Monte Carlo data the measured one. V(r) 0.29

4 Hybrid Phase 4-d Yukawa

a 0.288 0.2 0.288 4-d CoulombMC data 4d Yukawa fit 0.286 0.2 5d Yukawa fit dimensional reduction 5-d Coulomb4d Coulomb fit 5d Coulomb fit 0 0.284 ⇒ 1.5 2 0.284 2.5 5-d Yukawa 3

via localization 0.282 3 4 5 6 7 8 9 10 11 1.5 2.0 3 4 5 2.56 7r 8 9 10 113.0 β4 r/a4 [ Alberti, Irges, FK and Moir, 1506.06035 F. Knechtli, Higgs boson from 5d ] 10/14 Motivation Orbifold Model Phase diagram Spectrum Dimensional Reduction LCPs Conclusions Gauge-Higgs Uniﬁcation

Gauge-Higgs Uniﬁcation on the Orbifold Does it work?

Requirements of a viable model

it should reproduce the right physics. the ﬁfth dimension should be ”hidden”. the physics should be cut-off independent. Search for Lines of Constant Physics →

F. Knechtli, Higgs boson from 5d 11/14 Motivation Orbifold Model Phase diagram Spectrum Dimensional Reduction LCPs Conclusions Effective theory Non-renormalizability

I Gauge theories in ﬁve dimensions are perturbatively non-renormalizable.

I On the lattice so far the only known continuum limit a4 0 is perturbative and thus trivial. → −1 I Effective theory for energies E a4 : is there a range where cut-off effects are small? LPCP

Theory has three parameters: β4, β5 and N5 Lines of Constant Physics (LCP): keep two dimensionless quantities ﬁxed, e.g. ρ = mH /mZ and F1 = mH R and vary a4mZ . To begin: for several N5 = 4, 6, 8 construct Lines of Partially Constant Physics (LPCP) deﬁned by ρ = 1.15. F. Knechtli, Higgs boson from 5d 12/14 Motivation Orbifold Model Phase diagram Spectrum Dimensional Reduction LCPs Conclusions Lines of Partially Constant Physics

N = 4 5 Z N = 6 5 5 /m ′ 4.5

N = 8 H 5 1.3 m 4 ≡ Z 3 3.5 ρ 1.2 /m H Z m 1.1 2.5 /m ≡ ′

Z 2.3 ρ

1 m 2.1 ≡ 1.9 2 ρ 2.6 2.8 3 3.2 3.4 3.6 2.6 2.8 3 3.2 3.4 3.6 β4 β4

I On LPCPs a4mZ mainly depends on N5. It diminishes by 20%, going from 0.155 at N5 = 4 to 0.125 at N5 = 8. 0 0 I Energies of excited Higgs (H ) and excited gauge (Z ) bosons are approximately constant, ρ3 = mH0 /mZ 4.5 and ρ2 = mZ0 /mZ 2.3. ≈ ≈ [ Alberti, Irges, FK and Moir, arXiv:1609.07004 ] F. Knechtli, Higgs boson from 5d 13/14 Motivation Orbifold Model Phase diagram Spectrum Dimensional Reduction LCPs Conclusions Conclusions

Non-perturbative Gauge-Higgs Uniﬁcation

I SU(2) pure gauge theory on a 5d orbifold has a Higgs phase with the Higgs mechanism realized as a quantum and bosonic effect.

I A Standard-Model like spectrum can be reproduced.

I Localization on the 4d boundaries is observed.

I Cut-off effects appears to be small. Excited state energies are 2–5 times larger than the ground states. Outlook

I Pursue the construction of LCPs.

I Study connection to 4d Abelian Higgs model.

I Larger gauge group, warping, fermions, ...

Review on lattice works on extra dimensions [ FK and Rinaldi, arXiv:1605.04341 ]. F. Knechtli, Higgs boson from 5d 14/14