Rethinking Flavor Physics

Transcription

1 Rethinking Flavor Physics René Sondenheimer FSU Jena & L. Egger, A. Maas arxiv: Cold Quantum Coffee, Heidelberg 30th of May 2017

2 LHC works remarkably well Higgs discovery completed SM 2

3 LHC works remarkably well Higgs discovery completed SM No direct signal of BSM physics 2

4 The Standard Model in a nutshell Ɣ 0 e MeV Gauge theory: U(1) 3

5 The Standard Model in a nutshell u +2/3 g 2.3 MeV -1/3 d 4.8 MeV Ɣ 0 0 e MeV Gauge theory: SU(3)C x U(1) 4

6 The Standard Model in a nutshell u +2/3 g 2.3 MeV -1/3 d 4.8 MeV Ɣ 0 0 e -1 W ± MeV 80.4 GeV ν e <2eV Z 91.2 GeV Gauge theory: SU(3)C x SU(2)L x U(1)Y 5

7 The Standard Model in a nutshell +2/3 +2/3 +2/3 u c t g 2.3 MeV GeV GeV -1/3-1/3-1/3 d s b 4.8 MeV 95 MeV 4.18 GeV Ɣ e μ τ -1 W ± MeV MeV 1.78 GeV 80.4 GeV ν e ν μ ν τ Z <2eV <0.17 MeV <15.5 MeV 91.2 GeV Gauge theory: SU(3)C x SU(2)L x U(1)Y 6

8 The Standard Model in a nutshell +2/3 +2/3 +2/3 u c t g 2.3 MeV GeV GeV -1/3-1/3-1/3 d s b 4.8 MeV 95 MeV 4.18 GeV Ɣ MeV MeV 1.78 GeV -1 e μ τ -1 W 80.4 GeV ±1 H ν e ν μ ν τ Z 125 GeV <2eV <0.17 MeV <15.5 MeV 91.2 GeV Gauge theory: SU(3)C x SU(2)L x U(1)Y 7

9 Quantum ChromoDynamics Elementary fields: Quarks and Gluons L = 1 4 Giµ G i µ + q a i /D ab q b + m q q a q a 8

10 Quantum ChromoDynamics Elementary fields: Quarks and Gluons L = 1 4 Giµ G i µ + q a i /D ab q b + m q q a q a Observable states: hadrons q q q q q g g g q q q q q q q q q q q Mesons Baryons Glueballs? Hybrids? Tetraquarks? Pentaquarks? 8

11 Quantum ChromoDynamics Elementary fields: Quarks and Gluons L = 1 4 Giµ G i µ + q a i /D ab q b + m q q a q a Observable states: hadrons q q q q q g g g q q q q q q q q q q q Mesons Baryons Glueballs? Hybrids? Tetraquarks? Pentaquarks? Can we observe a single quark? No. 8

12 Quantum ChromoDynamics Elementary fields: Quarks and Gluons L = 1 4 Giµ G i µ + q a i /D ab q b + m q q a q a Observable states: hadrons q q q q q g g g q q q q q q q q q q q Mesons Baryons Glueballs? Hybrids? Tetraquarks? Pentaquarks? Can we observe a single quark? No. Gauge invariance! 8

13 Higgs-W Subsector Elementary fields: Higgs and W L = 1 4 W iµ W i µ +(D µ ) a (D µ ) a U( a a ) 9

14 Higgs-W Subsector Elementary fields: Higgs and W L = 1 4 W iµ W i µ +(D µ ) a (D µ ) a U( a a ) U L Higgs potential: U = µ ( ) 2 f 9

15 Higgs-W Subsector Elementary fields: Higgs and W L = 1 4 W iµ W i µ +(D µ ) a (D µ ) a U( a a ) U L Higgs potential: U = µ ( ) 2 Expansion around vev: (x) = 1 ' p + (x) 2 v + h(x)+' 0 (x) f 9

16 Higgs-W Subsector Elementary fields: Higgs and W L = 1 4 W iµ W i µ +(D µ ) a (D µ ) a U( a a ) U L Higgs potential: U = µ ( ) 2 Expansion around vev: (x) = 1 ' p + (x) 2 v + h(x)+' 0 (x) f Mass for the W s from the Higgs kinetic term (D µ ) D µ = 1 2 g 2 v 2 4 W i µw iµ

17 Higgs-W Subsector BUT! Elementary states are not gauge invariant Cannot be observable Gauge-invariant states are composite e.g., Higgs-Higgs, W-W, Higgs-W,. W Bound states are not asymptotic states in perturbation theory Why does perturbation theory works at all? Higgs condensate exist only in some gauges Local symmetry breaking vs Elitzur s theorem Elitzur 81 Lee et al 72, Osterwalder&Seiler 77 10

18 Outline Introduction FMS mechanism Implications for Fermions Generations (FMS beyond the standard model) Summary 11

19 Fröhlich-Morchio-Strocchi mechanism Fröhlich et al 80, 81 12

20 Fröhlich-Morchio-Strocchi mechanism Fröhlich et al 80, Construct a gauge-invariante operator 12

21 Fröhlich-Morchio-Strocchi mechanism Fröhlich et al 80, Construct a gauge-invariante operator O(x) =( )(x) 12

22 Fröhlich-Morchio-Strocchi mechanism Fröhlich et al 80, Construct a gauge-invariante operator O(x) =( )(x) 2. Expand Higgs field in correlator in fluctuations around the vev 12

23 Fröhlich-Morchio-Strocchi mechanism Fröhlich et al 80, Construct a gauge-invariante operator O(x) =( )(x) 2. Expand Higgs field in correlator in fluctuations around the vev = v + ' ho(x)o(y)i = const. + v 2 hh(x)h(y)i + v 3 h'i + vh' 3 i + h' 4 i 12

24 Fröhlich-Morchio-Strocchi mechanism Fröhlich et al 80, Construct a gauge-invariante operator O(x) =( )(x) 2. Expand Higgs field in correlator in fluctuations around the vev = v + ' ho(x)o(y)i = const. + v 2 hh(x)h(y)i + v 3 h'i + vh' 3 i + h' 4 i 3. Perform standard perturbation theory on the right-hand side 12

25 Fröhlich-Morchio-Strocchi mechanism Fröhlich et al 80, Construct a gauge-invariante operator O(x) =( )(x) 2. Expand Higgs field in correlator in fluctuations around the vev = v + ' ho(x)o(y)i = const. + v 2 hh(x)h(y)i + v 3 h'i + vh' 3 i + h' 4 i 3. Perform standard perturbation theory on the right-hand side ho(x)o(y)i = v 2 hh(x)h(y)i tl + hh(x)h(y)i 2 tl + O(g 2, ) 12

26 Fröhlich-Morchio-Strocchi mechanism Fröhlich et al 80, Construct a gauge-invariante operator O(x) =( )(x) 2. Expand Higgs field in correlator in fluctuations around the vev = v + ' ho(x)o(y)i = const. + v 2 hh(x)h(y)i + v 3 h'i + vh' 3 i + h' 4 i 3. Perform standard perturbation theory on the right-hand side ho(x)o(y)i = v 2 hh(x)h(y)i tl + hh(x)h(y)i 2 tl + O(g 2, ) 4. Compare poles on both sides 12

27 Fröhlich-Morchio-Strocchi mechanism Fröhlich et al 80, Construct a gauge-invariante operator O(x) =( )(x) 2. Expand Higgs field in correlator in fluctuations around the vev = v + ' ho(x)o(y)i = const. + v 2 hh(x)h(y)i + v 3 h'i + vh' 3 i + h' 4 i 3. Perform standard perturbation theory on the right-hand side ho(x)o(y)i = v 2 hh(x)h(y)i tl + hh(x)h(y)i 2 tl + O(g 2, ) 4. Compare poles on both sides Confirmed on the lattice for SU(2)-Higgs theory [Maas 12] 12

28 Fröhlich-Morchio-Strocchi mechanism 13

29 Fröhlich-Morchio-Strocchi mechanism W-Higgs sector of the standard model L = 1 4 W µ W i iµ +(D µ ) D µ U( ) Local SU(2) L Symmetry 13

30 Fröhlich-Morchio-Strocchi mechanism W-Higgs sector of the standard model L = 1 4 W i µ W iµ +(D µ ) D µ U( ) Local SU(2) L Symmetry W µ! UW µ U 1 i g (@ µu)u 1,! U 13

31 Fröhlich-Morchio-Strocchi mechanism W-Higgs sector of the standard model L = 1 4 W i µ W iµ +(D µ ) D µ U( ) Local SU(2) L Symmetry W µ! UW µ U 1 i g (@ µu)u 1,! U Additional global SU(2) cust. symmetry 13

32 Fröhlich-Morchio-Strocchi mechanism W-Higgs sector of the standard model L = 1 4 W i µ W iµ +(D µ ) D µ U( ) Local SU(2) L Symmetry W µ! UW µ U 1 i g (@ µu)u 1,! U Additional global SU(2) cust. symmetry W µ! W µ,! V + W 13

33 Fröhlich-Morchio-Strocchi mechanism W-Higgs sector of the standard model L = 1 4 W i µ W iµ +(D µ ) D µ U( ) Local SU(2) L Symmetry W µ! UW µ U 1 i g (@ µu)u 1,! U Additional global SU(2) cust. symmetry W µ! W µ,! V + W Convenient to rewrite Higgs-Lagrangian 13

34 Fröhlich-Morchio-Strocchi mechanism W-Higgs sector of the standard model L = 1 4 W i µ W iµ +(D µ ) D µ U( ) Local SU(2) L Symmetry W µ! UW µ U 1 i g (@ µu)u 1,! U Additional global SU(2) cust. symmetry W µ! W µ,! V + W Convenient to rewrite Higgs-Lagrangian L =Tr[@ µ µ X] U(TrX X) 13

35 Fröhlich-Morchio-Strocchi mechanism W-Higgs sector of the standard model L = 1 4 W i µ W iµ +(D µ ) D µ U( ) Local SU(2) L Symmetry W µ! UW µ U 1 i g (@ µu)u 1,! U Additional global SU(2) cust. symmetry W µ! W µ,! V + W Convenient to rewrite Higgs-Lagrangian L =Tr[@ µ µ X] U(TrX X) where X = = = v 0 0 v + O(')

36 Fröhlich-Morchio-Strocchi mechanism 14

37 Fröhlich-Morchio-Strocchi mechanism Full symmetry acting on the Higgs field: SU(2) L x SU(2)cust. 14

38 Fröhlich-Morchio-Strocchi mechanism Full symmetry acting on the Higgs field: SU(2) L x SU(2)cust. X! U L XU cust 14

39 Fröhlich-Morchio-Strocchi mechanism Full symmetry acting on the Higgs field: SU(2) L x SU(2)cust. X! U L XU cust W mass: bound state operator in 1- channel 14

40 Fröhlich-Morchio-Strocchi mechanism Full symmetry acting on the Higgs field: SU(2) L x SU(2)cust. X! U L XU cust W mass: bound state operator in 1- channel Oĩµ =itr( ĩx D µ X) 14

41 Fröhlich-Morchio-Strocchi mechanism Full symmetry acting on the Higgs field: SU(2) L x SU(2)cust. X! U L XU cust W mass: bound state operator in 1- channel Expand Higgs field: Oĩµ =itr( ĩx D µ X) 14

42 Fröhlich-Morchio-Strocchi mechanism Full symmetry acting on the Higgs field: SU(2) L x SU(2)cust. X! U L XU cust W mass: bound state operator in 1- channel Expand Higgs field: Oĩµ =itr( ĩx D µ X) hoĩµ(x)o j (y)i = g 2 v 4 hw i µ(x)w j (y)i iĩ j j + O(' 2 ) 14

43 Fröhlich-Morchio-Strocchi mechanism Full symmetry acting on the Higgs field: SU(2) L x SU(2)cust. X! U L XU cust W mass: bound state operator in 1- channel Expand Higgs field: Oĩµ =itr( ĩx D µ X) hoĩµ(x)o j (y)i = g 2 v 4 hw i µ(x)w j (y)i iĩ j j + O(' 2 ) Pole of bound state the same as for the elementary fields 14

44 Fröhlich-Morchio-Strocchi mechanism Full symmetry acting on the Higgs field: SU(2) L x SU(2)cust. X! U L XU cust W mass: bound state operator in 1- channel Expand Higgs field: Oĩµ =itr( ĩx D µ X) hoĩµ(x)o j (y)i = g 2 v 4 hw i µ(x)w j (y)i iĩ j j + O(' 2 ) Pole of bound state the same as for the elementary fields Mapping of local to global multiplets 14

45 Fermions Fröhlich et al 80 Egger,Maas,RS 17 15

46 Fermions Fröhlich et al 80 Egger,Maas,RS 17 Flavor = weak gauge charge in a given generation 15

47 Fermions Fröhlich et al 80 Egger,Maas,RS 17 Flavor = weak gauge charge in a given generation Necessary to construct suitable gauge-invariant state which emulate elementary fermions 15

48 Fermions Fröhlich et al 80 Egger,Maas,RS 17 Flavor = weak gauge charge in a given generation Necessary to construct suitable gauge-invariant state which emulate elementary fermions O g (x) =X (x)f g (x) 15

49 Fermions Fröhlich et al 80 Egger,Maas,RS 17 Flavor = weak gauge charge in a given generation Necessary to construct suitable gauge-invariant state which emulate elementary fermions O g (x) =X (x)f g (x) State remains custodial doublet 15

50 Fermions Fröhlich et al 80 Egger,Maas,RS 17 Flavor = weak gauge charge in a given generation Necessary to construct suitable gauge-invariant state which emulate elementary fermions O g (x) =X (x)f g (x) State remains custodial doublet E.g., (left-handed) electron and neutrino 15

51 Fermions Fröhlich et al 80 Egger,Maas,RS 17 Flavor = weak gauge charge in a given generation Necessary to construct suitable gauge-invariant state which emulate elementary fermions O g (x) =X (x)f g (x) State remains custodial doublet E.g., (left-handed) electron and neutrino O e = X = e e 2e = v e + O(') 15

52 Fermions Fröhlich et al 80 Egger,Maas,RS 17 Flavor = weak gauge charge in a given generation Necessary to construct suitable gauge-invariant state which emulate elementary fermions O g (x) =X (x)f g (x) State remains custodial doublet E.g., (left-handed) electron and neutrino O e = X = e e 2e = v e + O(') Ordinary states remerge when applying FMS 15

53 Fermions - Quarks Egger,Maas,RS 17 16

54 Fermions - Quarks O ud = X u d = 2u 1u + 1d 2d = v u d + O(') Egger,Maas,RS 17 16

55 Fermions - Quarks O ud = X u = d 2u 1d u 1u + = v + O(') 2d d Egger,Maas,RS 17 Quark sector different from lepton sector, bound in hadrons 16

56 Fermions - Quarks O ud = X u d = 2u 1u + 1d 2d Quark sector different from lepton sector, bound in hadrons Hadrons singlets of strong interaction, not necessarily of weak interaction = v u d + O(') Egger,Maas,RS 17 16

57 Fermions - Quarks O ud = X u d Quark sector different from lepton sector, bound in hadrons Hadrons singlets of strong interaction, not necessarily of weak interaction Obvious for baryons (strong indices suppressed) = 2u 1u + 1d 2d = v u d + O(') Egger,Maas,RS 17 16

58 Fermions - Quarks O ud = X u d Quark sector different from lepton sector, bound in hadrons Hadrons singlets of strong interaction, not necessarily of weak interaction Obvious for baryons (strong indices suppressed) = 2u 1u + 1d 2d = v u d + O(') Egger,Maas,RS 17 c ijkl q i q j q k X ĩl or q i q j q k X ĩi X jj X kk 16

59 Fermions - Quarks O ud = X u d Quark sector different from lepton sector, bound in hadrons Hadrons singlets of strong interaction, not necessarily of weak interaction Obvious for baryons (strong indices suppressed) = 2u 1u + 1d 2d = v u d + O(') Egger,Maas,RS 17 c ijkl q i q j q k X ĩl or q i q j q k X ĩi X jj X kk c ijkl = a 1 ij kl + a 2 ik jl + a 3 jk il and ĩ =1 16

60 Fermions - Quarks O ud = X u d Quark sector different from lepton sector, bound in hadrons Hadrons singlets of strong interaction, not necessarily of weak interaction Obvious for baryons (strong indices suppressed) = 2u 1u + 1d 2d = v u d + O(') Egger,Maas,RS 17 c ijkl q i q j q k X ĩl or q i q j q k X ĩi X jj X kk Proton: c ijkl = a 1 ij kl + a 2 ik jl + a 3 jk il and ĩ =1 16

61 Fermions - Quarks O ud = X u d Quark sector different from lepton sector, bound in hadrons Hadrons singlets of strong interaction, not necessarily of weak interaction Obvious for baryons (strong indices suppressed) = 2u 1u + 1d 2d = v u d + O(') Egger,Maas,RS 17 c ijkl q i q j q k X ĩl or q i q j q k X ĩi X jj X kk Proton: c ijkl = a 1 ij kl + a 2 ik jl + a 3 jk il and ĩ =1 Some mesons are weak-gauge singlets, e.g., 16

62 Fermions - Quarks O ud = X u d Quark sector different from lepton sector, bound in hadrons Hadrons singlets of strong interaction, not necessarily of weak interaction Obvious for baryons (strong indices suppressed) = 2u 1u + 1d 2d = v u d + O(') Egger,Maas,RS 17 c ijkl q i q j q k X ĩl or q i q j q k X ĩi X jj X kk Proton: c ijkl = a 1 ij kl + a 2 ik jl + a 3 jk il and ĩ =1 Some mesons are weak-gauge singlets, e.g.,! meson (ūu + dd) 16

63 Fermions - Quarks O ud = X u d Quark sector different from lepton sector, bound in hadrons Hadrons singlets of strong interaction, not necessarily of weak interaction Obvious for baryons (strong indices suppressed) = 2u 1u + 1d 2d = v u d + O(') Egger,Maas,RS 17 c ijkl q i q j q k X ĩl or q i q j q k X ĩi X jj X kk Proton: c ijkl = a 1 ij kl + a 2 ik jl + a 3 jk il and ĩ =1 Some mesons are weak-gauge singlets, e.g.,! meson (ūu + dd) Not true for all mesons, e.g., pions 16

64 Fermions - Quarks O ud = X u d Quark sector different from lepton sector, bound in hadrons Hadrons singlets of strong interaction, not necessarily of weak interaction Obvious for baryons (strong indices suppressed) = 2u 1u + 1d 2d = v u d + O(') Egger,Maas,RS 17 c ijkl q i q j q k X ĩl or q i q j q k X ĩi X jj X kk Proton: c ijkl = a 1 ij kl + a 2 ik jl + a 3 jk il and ĩ =1 Some mesons are weak-gauge singlets, e.g.,! meson (ūu + dd) Not true for all mesons, e.g., pions 16 + : Ō ud 2 O ud 1 ( du)

65 Generations as excitation spectra (speculation!) Egger,Maas,RS 17 17

66 Generations as excitation spectra (speculation!) Egger,Maas,RS 17 Flavor identity from combination of generation and weak isospin 17

67 Generations as excitation spectra (speculation!) Egger,Maas,RS 17 Flavor identity from combination of generation and weak isospin Difference: Yukawa and weak interactions cannot be simultaneously diagonal in generation space 17

68 Generations as excitation spectra (speculation!) Egger,Maas,RS 17 Flavor identity from combination of generation and weak isospin Difference: Yukawa and weak interactions cannot be simultaneously diagonal in generation space Flavor distinguishes between inter- and intrageneration effects 17

69 Generations as excitation spectra (speculation!) Egger,Maas,RS 17 Flavor identity from combination of generation and weak isospin Difference: Yukawa and weak interactions cannot be simultaneously diagonal in generation space Flavor distinguishes between inter- and intrageneration effects Speculation(!): 3 generations could be internal excitations of a single generation 17

70 Generations as excitation spectra (speculation!) Egger,Maas,RS 17 Flavor identity from combination of generation and weak isospin Difference: Yukawa and weak interactions cannot be simultaneously diagonal in generation space Flavor distinguishes between inter- and intrageneration effects Speculation(!): 3 generations could be internal excitations of a single generation SM effective theory of excitation spectrum 17

71 Generations as excitation spectra (speculation!) Egger,Maas,RS 17 Flavor identity from combination of generation and weak isospin Difference: Yukawa and weak interactions cannot be simultaneously diagonal in generation space Flavor distinguishes between inter- and intrageneration effects Speculation(!): 3 generations could be internal excitations of a single generation SM effective theory of excitation spectrum X Q E = T T Q E Q E = 2u E 2d E + 1 d E 1u E 17

72 Generations as excitation spectra (speculation!) Egger,Maas,RS 17 Flavor identity from combination of generation and weak isospin Difference: Yukawa and weak interactions cannot be simultaneously diagonal in generation space Flavor distinguishes between inter- and intrageneration effects Speculation(!): 3 generations could be internal excitations of a single generation SM effective theory of excitation spectrum Ground state: X Q E = T T Q E Q E = 2u E 2d E + 1 d E 1u E 17

73 Generations as excitation spectra (speculation!) Egger,Maas,RS 17 Flavor identity from combination of generation and weak isospin Difference: Yukawa and weak interactions cannot be simultaneously diagonal in generation space Flavor distinguishes between inter- and intrageneration effects Speculation(!): 3 generations could be internal excitations of a single generation SM effective theory of excitation spectrum X Q E = T T Q E Q E = 2u E 2d E + 1 d E 1u E Ground state: d =( Q E ) g 17

74 Generations as excitation spectra (speculation!) Egger,Maas,RS 17 Flavor identity from combination of generation and weak isospin Difference: Yukawa and weak interactions cannot be simultaneously diagonal in generation space Flavor distinguishes between inter- and intrageneration effects Speculation(!): 3 generations could be internal excitations of a single generation SM effective theory of excitation spectrum X Q E = T T Q E Q E = 2u E 2d E + 1 d E 1u E Ground state: d =( Q E ) g higher excitations: 17

75 Generations as excitation spectra (speculation!) Egger,Maas,RS 17 Flavor identity from combination of generation and weak isospin Difference: Yukawa and weak interactions cannot be simultaneously diagonal in generation space Flavor distinguishes between inter- and intrageneration effects Speculation(!): 3 generations could be internal excitations of a single generation SM effective theory of excitation spectrum X Q E = T T Q E Q E = 2u E 2d E + 1 d E 1u E Ground state: higher excitations: d =( Q E ) g s =( Q E ), b =( Q E ) 17

76 Gauge-invariant perturbation theory Egger,Maas,RS 17 Lepton collision as probing experiment of the bound state structure M = ho e 2 (p 1 )Ō e 2 (p 2 )O f (q 1 )Ōf (q 2 )i v 4 he + e ffi + he + e ihhh ffi +... Final state μ b t 1.00 dσ Full dσ Perturbative s 18

77 BSM - Grand Unified theories Maas,Törek 16 Maas,Törek,RS, in preparation 19

78 BSM - Grand Unified theories Maas,Törek 16 Maas,Törek,RS, in preparation SU(N) gauge theory + Higgs in fundamental representation Perturbative mass spectrum: 2N - 1 massive gauge bosons (N-1) 2-1 massless gauge bosons 1 massive scalar Higgs boson 19

79 BSM - Grand Unified theories Maas,Törek 16 Maas,Törek,RS, in preparation SU(N) gauge theory + Higgs in fundamental representation Perturbative mass spectrum: 2N - 1 massive gauge bosons (N-1) 2-1 massless gauge bosons 1 massive scalar Higgs boson Local and global symmetry group do not match for N > 2 19

80 BSM - Grand Unified theories Maas,Törek 16 Maas,Törek,RS, in preparation SU(N) gauge theory + Higgs in fundamental representation Perturbative mass spectrum: 2N - 1 massive gauge bosons (N-1) 2-1 massless gauge bosons 1 massive scalar Higgs boson Local and global symmetry group do not match for N > 2 1 neutral bound state in 0+ channel = Higgs 19

81 BSM - Grand Unified theories Maas,Törek 16 Maas,Törek,RS, in preparation SU(N) gauge theory + Higgs in fundamental representation Perturbative mass spectrum: 2N - 1 massive gauge bosons (N-1) 2-1 massless gauge bosons 1 massive scalar Higgs boson Local and global symmetry group do not match for N > 2 1 neutral bound state in 0+ channel = Higgs Only 1 bound state in 1- channel (U(1) singlet) i D µ 19

82 BSM - Grand Unified theories Maas,Törek 16 Maas,Törek,RS, in preparation SU(N) gauge theory + Higgs in fundamental representation Perturbative mass spectrum: 2N - 1 massive gauge bosons (N-1) 2-1 massless gauge bosons 1 massive scalar Higgs boson Local and global symmetry group do not match for N > 2 1 neutral bound state in 0+ channel = Higgs Only 1 bound state in 1- channel (U(1) singlet) i D µ 1- states with open U(1) quantum number? abc a (D µ ) b (D D ) c (for SU(3)) 19

83 Summary Observable spectrum must be gauge invariant Non-Abelian gauge theory: composite operator FMS mechanism provides a mapping of the local to the global multiplets Gauge-invariant perturbation theory as a tool BSM model building may be affected Same results in leading order for the standard model Verification requieres non-perturbative methods 20