The physical spectrum of theories with a Brout-Englert-Higgs effect
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1 The physical spectrum of theories with a Brout-Englert-Higgs effect Pascal Törek with Axel Maas and René Sondenheimer University of Graz Alps 2018, Obergurgl, 18 th of April, 2018 [ and ]
2 Motivation Physical observable particles need to be manifestly gauge invariant Elementary fields of gauge theories: gauge variant 2 / 14
3 Motivation Physical observable particles need to be manifestly gauge invariant Elementary fields of gauge theories: gauge variant Not obvious that W/Z, Higgs, fermions, etc., are physical particles from theoretical p.o.v. 2 / 14
4 Motivation Physical observable particles need to be manifestly gauge invariant Elementary fields of gauge theories: gauge variant Not obvious that W/Z, Higgs, fermions, etc., are physical particles from theoretical p.o.v. Does not matter in the SM Can matter in BSM theories 2 / 14
5 Motivation Physical observable particles need to be manifestly gauge invariant Elementary fields of gauge theories: gauge variant Not obvious that W/Z, Higgs, fermions, etc., are physical particles from theoretical p.o.v. Does not matter in the SM Can matter in BSM theories Problems can be treated using gauge-invariant perturbation theory [Maas, Review] 2 / 14
6 Weak-Higgs sector of SM - Basics Consider bosonic weak-higgs sector of SM L = 1 4 W a µνw a µν + ( D µ φ ) ( D µ φ ) V ( φ φ ) Full symmetry: SU(2) gauge SU(2) custodial 3 / 14
7 Weak-Higgs sector of SM - Basics Consider bosonic weak-higgs sector of SM L = 1 4 W a µνw a µν + ( D µ φ ) ( D µ φ ) V ( φ φ ) Full symmetry: SU(2) gauge SU(2) custodial Standard approach Minimize action classically: Higgs vev φ φ = v 2 3 / 14
8 Weak-Higgs sector of SM - Basics Consider bosonic weak-higgs sector of SM L = 1 4 W a µνw a µν + ( D µ φ ) ( D µ φ ) V ( φ φ ) Full symmetry: SU(2) gauge SU(2) custodial Standard approach Minimize action classically: Higgs vev φ φ = v 2 Perform gauge transformation such that φ(x) = v n + ϕ(x) = 1 ( ) ϕ1 (x) + i ϕ 2 (x) 2 2 v + h(x) + i ϕ 3 (x) ( 0, φ = v) 3 / 14
9 Weak-Higgs sector of SM - Basics Consider bosonic weak-higgs sector of SM L = 1 4 W a µνw a µν + ( D µ φ ) ( D µ φ ) V ( φ φ ) Full symmetry: SU(2) gauge SU(2) custodial Standard approach Minimize action classically: Higgs vev φ φ = v 2 Perform gauge transformation such that φ(x) = v n + ϕ(x) = 1 ( ) ϕ1 (x) + i ϕ 2 (x) 2 2 v + h(x) + i ϕ 3 (x) ( 0, φ = v) Masses of Higgs, W/Z, depend on vev 3 / 14
10 Weak-Higgs sector of SM - Basics Consider bosonic weak-higgs sector of SM L = 1 4 W a µνw a µν + ( D µ φ ) ( D µ φ ) V ( φ φ ) Full symmetry: SU(2) gauge SU(2) custodial Standard approach Minimize action classically: Higgs vev φ φ = v 2 Perform gauge transformation such that φ(x) = v n + ϕ(x) = 1 ( ) ϕ1 (x) + i ϕ 2 (x) 2 2 v + h(x) + i ϕ 3 (x) ( 0, φ = v) Masses of Higgs, W/Z, depend on vev Perform PT (small fluctuations ϕ) 3 / 14
11 Weak-Higgs sector of SM - Spectrum 0mass Perturbation theory scalar h vector W a µ Custodial singlets 4 / 14
12 Weak-Higgs sector of SM - Spectrum 0mass Perturbation theory scalar h vector W a µ Custodial singlets Gauge transformation for vev: Gauge choice Spontaneous gauge symmetry breaking There are gauges where φ = 0 PT not sensible Symmetry is not manifest (hidden) 4 / 14
13 Physical states Elementary fields (h, W,... ) depend on the gauge Cannot be observable 5 / 14
14 Physical states Elementary fields (h, W,... ) depend on the gauge Cannot be observable Gauge-invariant states in gauge theories are composite objects: Higgs-Higgs φ φ W-Ball W W Higgs-Higgs-W W et cetera... φ φ 5 / 14
15 Physical states Elementary fields (h, W,... ) depend on the gauge Cannot be observable Gauge-invariant states in gauge theories are composite objects: Higgs-Higgs φ φ W-Ball W W Higgs-Higgs-W W et cetera... φ φ Why does perturbation theory work so well? What is the mass spectrum? 5 / 14
16 Weak-Higgs sector of SM - Spectrum Lattice spectroscopy Spectrum of bound states [Maas, MPL A28 (2013) / Maas and Mufti, JHEP (2014)] 6 / 14
17 Weak-Higgs sector of SM - Spectrum Lattice spectroscopy Spectrum of bound states 0mass [Maas, MPL A28 (2013) / Maas and Mufti, JHEP (2014)] Perturbation theory scalar h vector W a µ Custodial singlets Gauge-invariant scalar vector singlet triplet Masses of bound states = elementary fields 6 / 14
18 Weak-Higgs sector of SM - Spectrum Lattice spectroscopy Spectrum of bound states 0mass [Maas, MPL A28 (2013) / Maas and Mufti, JHEP (2014)] Perturbation theory scalar h vector W a µ Custodial singlets Gauge-invariant scalar vector singlet triplet Masses of bound states = elementary fields This is not a coincidence! [Fröhlich et al., PL B97 (1980) and NP B190 (1981)] 6 / 14
19 Gauge-invariant perturbation theory I Gauge-invariant operator in 0 + channel: O(x) = ( φ φ ) (x) 7 / 14
20 Gauge-invariant perturbation theory I Gauge-invariant operator in 0 + channel: O(x) = ( φ φ ) (x) Fix to a gauge with non-vanishing vev φ = v 2 n 7 / 14
21 Gauge-invariant perturbation theory I Gauge-invariant operator in 0 + channel: O(x) = ( φ φ ) (x) Fix to a gauge with non-vanishing vev φ = v 2 n Expand correlator in Higgs fluctuations ϕ (FMS mechanism) φ= v n+ϕ 2 O(x)O(y) v 4 = h= 2Re[n φ] 4 + v 3 h(x) + h(y) + v 2 h(x)h(y) 2 + v 2 ( h(x) ϕ ϕ ) (y) + ( ϕ ϕ ) (x)h(y) + ( ϕ ϕ ) (x) ( ϕ ϕ ) (y) 2 7 / 14
22 Gauge-invariant perturbation theory I Gauge-invariant operator in 0 + channel: O(x) = ( φ φ ) (x) Fix to a gauge with non-vanishing vev φ = v 2 n Expand correlator in Higgs fluctuations ϕ (FMS mechanism) φ= v n+ϕ 2 O(x)O(y) v 4 = h= 2Re[n φ] 4 + v 3 h(x) + h(y) + v 2 h(x)h(y) 2 + v 2 ( h(x) ϕ ϕ ) (y) + ( ϕ ϕ ) (x)h(y) + ( ϕ ϕ ) (x) ( ϕ ϕ ) (y) 2 Exact identity 7 / 14
23 Gauge-invariant perturbation theory I Gauge-invariant operator in 0 + channel: O(x) = ( φ φ ) (x) Fix to a gauge with non-vanishing vev φ = v 2 n Expand correlator in Higgs fluctuations ϕ (FMS mechanism) φ= v n+ϕ 2 O(x)O(y) v 4 = h= 2Re[n φ] 4 + v 3 h(x) + h(y) + v 2 h(x)h(y) 2 + v 2 ( h(x) ϕ ϕ ) (y) + ( ϕ ϕ ) (x)h(y) + ( ϕ ϕ ) (x) ( ϕ ϕ ) (y) 2 Exact identity Sum on r.h.s. is gauge-invariant but each term individually is gauge-variant 7 / 14
24 Gauge-invariant perturbation theory II Perform standard perturbation theory on r.h.s.: O(x)O(y) = v v 2 h(x)h(y) tl + h(x)h(y) 2 tl + O(ϕ3, g, λ) 8 / 14
25 Gauge-invariant perturbation theory II Perform standard perturbation theory on r.h.s.: O(x)O(y) = v v 2 h(x)h(y) tl + h(x)h(y) 2 tl + O(ϕ3, g, λ) Compare poles on both sides: States at tree-level Higgs mass and at twice this mass (scattering state) 8 / 14
26 Gauge-invariant perturbation theory II Perform standard perturbation theory on r.h.s.: O(x)O(y) = v v 2 h(x)h(y) tl + h(x)h(y) 2 tl + O(ϕ3, g, λ) Compare poles on both sides: States at tree-level Higgs mass and at twice this mass (scattering state) FMS mechanism + standard PT = GIPT 8 / 14
27 Gauge-invariant perturbation theory II Perform standard perturbation theory on r.h.s.: O(x)O(y) = v v 2 h(x)h(y) tl + h(x)h(y) 2 tl + O(ϕ3, g, λ) Compare poles on both sides: States at tree-level Higgs mass and at twice this mass (scattering state) FMS mechanism + standard PT = GIPT Similar procedure for the W bosons 8 / 14
28 Gauge-invariant perturbation theory II Perform standard perturbation theory on r.h.s.: O(x)O(y) = v v 2 h(x)h(y) tl + h(x)h(y) 2 tl + O(ϕ3, g, λ) Compare poles on both sides: States at tree-level Higgs mass and at twice this mass (scattering state) FMS mechanism + standard PT = GIPT Similar procedure for the W bosons Confirmed on the lattice [Maas, MPL A28 (2013) / Maas and Mufti, JHEP (2014)] 8 / 14
29 Status of the standard model Physical states are bound states Experimentally observable Description by gauge-invariant perturbation theory based on FMS mechanism Mostly the same as ordinary perturbation theory 9 / 14
30 Status of the standard model Physical states are bound states Experimentally observable Description by gauge-invariant perturbation theory based on FMS mechanism Mostly the same as ordinary perturbation theory Does not always work [Maas, MPL A28 (2013) / Maas and Mufti, JHEP (2014)] Fluctuations can invalidate the mechanism Local and global multiplet structure must fit 9 / 14
31 Status of the standard model Physical states are bound states Experimentally observable Description by gauge-invariant perturbation theory based on FMS mechanism Mostly the same as ordinary perturbation theory Does not always work [Maas, MPL A28 (2013) / Maas and Mufti, JHEP (2014)] Fluctuations can invalidate the mechanism Local and global multiplet structure must fit Has to be checked for BSM theories 9 / 14
32 SU(N) + fundamental scalar - Toy GUT [Maas, Sondenheimer and Törek, ] GUT inspired theories: Gauge group is larger than global symmetry group 10 / 14
33 SU(N) + fundamental scalar - Toy GUT [Maas, Sondenheimer and Törek, ] GUT inspired theories: Gauge group is larger than global symmetry group Same logic as in SM leads to a conflict 10 / 14
34 SU(N) + fundamental scalar - Toy GUT [Maas, Sondenheimer and Törek, ] GUT inspired theories: Gauge group is larger than global symmetry group Same logic as in SM leads to a conflict Consider SU(N > 2) gauge theory with one fundamental scalar φ Global symmetry: U(1) 10 / 14
35 SU(N) + fundamental scalar - Toy GUT [Maas, Sondenheimer and Törek, ] GUT inspired theories: Gauge group is larger than global symmetry group Same logic as in SM leads to a conflict Consider SU(N > 2) gauge theory with one fundamental scalar φ Global symmetry: U(1) Perturbative construction: SU(N) φ SU(N 1) 2(N 1) + 1 massive and N(N 2) massless gauge bosons 1 massive real scalar field 10 / 14
36 Physical spectrum of the toy GUT [Maas, Sondenheimer and Törek, ] Global U(1) symmetry gauge-invariant U(1)-singlet and non-singlet bound states 11 / 14
37 Physical spectrum of the toy GUT [Maas, Sondenheimer and Törek, ] Global U(1) symmetry gauge-invariant U(1)-singlet and non-singlet bound states U(1)-singlet channels Scalar : 1 state with mass of elementary scalar Vector: 1 state with mass of heaviest g.b. 11 / 14
38 Physical spectrum of the toy GUT [Maas, Sondenheimer and Törek, ] Global U(1) symmetry gauge-invariant U(1)-singlet and non-singlet bound states U(1)-singlet channels Scalar : 1 state with mass of elementary scalar Vector: 1 state with mass of heaviest g.b. U(1)-non-singlet channels Scalar and vector channels: 2 states each with ground-state masses = (N 1) perturbative lightest massive gauge boson 11 / 14
39 Physical spectrum of the toy GUT [Maas, Sondenheimer and Törek, ] Global U(1) symmetry gauge-invariant U(1)-singlet and non-singlet bound states U(1)-singlet channels Scalar : 1 state with mass of elementary scalar Vector: 1 state with mass of heaviest g.b. U(1)-non-singlet channels Scalar and vector channels: 2 states each with ground-state masses = (N 1) perturbative lightest massive gauge boson Focus on N = 3 and vector channel in the following 11 / 14
40 SU(3) + fundamental scalar - Toy GUT [Maas and Törek, PRD95, (2017), and ] Spectrum in the vector channel mass PT GIPT gauge variant U(1)-singlet GIPT U(1)-non-singlet 2m light g.b. m heavy g.b. m light g.b / 14
41 SU(3) + fundamental scalar - Toy GUT [Maas and Törek, PRD95, (2017), and ] 2m light g.b. Spectrum in the vector channel mass PT GIPT gauge variant U(1)-singlet - ( ) GIPT U(1)-non-singlet m heavy g.b. m light g.b. 0 ( ) - 12 / 14
42 SU(3) + fundamental scalar - Toy GUT [Maas and Törek, PRD95, (2017), and ] Spectrum in the vector channel mass PT GIPT gauge variant U(1)-singlet GIPT U(1)-non-singlet 2m light g.b. m heavy g.b. m light g.b. = λ () λ ( +) 0 / 12 / 14
43 SU(3) + fundamental scalar - Toy GUT [Maas and Törek, PRD95, (2017), and ] 2m light g.b. Spectrum in the vector channel mass PT GIPT gauge variant U(1)-singlet () = = = GIPT U(1)-non-singlet -- U(1) = ± m heavy g.b. m light g.b. 0 2x ( ) ( +) = = / 12 / 14
44 Implications for GUTs Qualitative disagreement to standard PT but good agreement to GIPT for SU(3) U(1) case 13 / 14
45 Implications for GUTs Qualitative disagreement to standard PT but good agreement to GIPT for SU(3) U(1) case Disagreement generic [Maas, Sondenheimer and Törek, and work in progress] Lattice support for SU(2) with an adjoint scalar [Lee and Shigemitsu, NP B263 (1986)] 13 / 14
46 Implications for GUTs Qualitative disagreement to standard PT but good agreement to GIPT for SU(3) U(1) case Disagreement generic [Maas, Sondenheimer and Törek, and work in progress] Lattice support for SU(2) with an adjoint scalar [Lee and Shigemitsu, NP B263 (1986)] Conventional GUTs unlikely to reproduce low-energy spectrum according to GIPT Larger custodial groups needed (?) 13 / 14
47 Conclusions Physical spectrum consists of gauge-invariant states 14 / 14
48 Conclusions Physical spectrum consists of gauge-invariant states Relation between physical states and elementary fields by FMS mechanism / GIPT 14 / 14
49 Conclusions Physical spectrum consists of gauge-invariant states Relation between physical states and elementary fields by FMS mechanism / GIPT Collections of verifications/supports of mechanism Gauge-Higgs sector of SM [Maas, MPL A28 (2013); Maas and Mufti, JHEP (2014)] SU(2) U(1) with Higgs (no direct verification) [Shrock, various publications (1980 s)] SU(2) with adjoint scalar (no direct verification) [Lee and Shigemitsu, NP B263 (1986)] SU(3) with fundamental scalar [Maas and Törek, PRD95, (2017)] 14 / 14
50 Conclusions Physical spectrum consists of gauge-invariant states Relation between physical states and elementary fields by FMS mechanism / GIPT Collections of verifications/supports of mechanism Gauge-Higgs sector of SM [Maas, MPL A28 (2013); Maas and Mufti, JHEP (2014)] SU(2) U(1) with Higgs (no direct verification) [Shrock, various publications (1980 s)] SU(2) with adjoint scalar (no direct verification) [Lee and Shigemitsu, NP B263 (1986)] SU(3) with fundamental scalar [Maas and Törek, PRD95, (2017)] Procedure can be used to build or rule out BSM theories (e.g. std. SU(5)-GUT construction) 14 / 14
51 Thank you!
52 GIPT works well! ( ) ( +) 16 / 14
53 SU(5) GUT - Bosonic sector Fundamental scalar ϕ and adjoint scalar σ Custodial group: U(1) Z 2 GUT-scale: w elementary spectrum gauge-invariant spectrum J P Field Mass Deg. (U(1), Z 2 ) Mass N.-l. Deg. 0 + h m h 1 (0, +) m h w 1 ϕ 1,...,6 m ϕ1,...,6 6 (0, ) m h w 1 σ 1,...,8 m σ 1,...,8 8 (±1, +) w w 1 σ 21,22,23 m σ 3 (±1, ) w w 1 σ 24 M σ 1 1 A µ m A = 0 1 (0, +) m A m Z 1 W µ ± m W 2 (0, ) m A m Z 1 Z µ m Z 1 (±1, +) w w 1 Aµ 9,...,14 m L 6 (±1, ) w w 1 A 15,...,20 µ M L 6 17 / 14
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