The Higgs Mechanism and the Higgs Particle

Transcription

1 The Higgs Mechanism and the Higgs Particle Heavy-Ion Seminar... or the Anderson-Higgs-Brout-Englert-Guralnik-Hagen-Kibble Mechanism Philip W. Anderson Peter W. Higgs Tom W. B. Gerald Carl R. François Robert Kibble Guralnik Hagen Englert Brout (pictures from: wikipedia.org) December 16, 2010 Institut für Kernphysik, TU Darmstadt Pascal Büscher 1

2 Contents Motivation Basics Electroweak Sector of the Standard Model Alternatives to the Standard Model Summary December 16, 2010 Institut für Kernphysik, TU Darmstadt Pascal Büscher 2

3 Motivation In field theory gauge bosons usually massless Why are those of the weak interaction massive? What generates the (bare) mass of quarks? in the Standard model: the Higgs mechanism A (so far) not measured particle is introduced What are the properties of this particle? Is the Higgs mechanism the only possibility? Experimental talk: Strategies for Higgs searches on Feb. 10 th December 16, 2010 Institut für Kernphysik, TU Darmstadt Pascal Büscher 4

4 Covariant Derivative local U(1) kinetic term for fermions: L ferm = ψiγ µ µ ψ local U(1) trafo: ψ ψ = e iλ(x) ψ(x) if symmetry obeyed L ferm = L ferm µ ψ µ ψ = e iλ ( i( µ Λ) + µ ) ψ define D µ µ iea(x) e: coupling constant, A(x): gauge field absorb µ Λ term in A(x) kinetic term for gauge field: L kin,gauge = [ ] 1 4 F µνf µν with F µν = 1 e Dµ, D ν = µ A ν ν A µ December 16, 2010 Institut für Kernphysik, TU Darmstadt Pascal Büscher 6

5 Covariant Derivative local SU(2) local SU(2) trafo: ψ ψ = e i 2 τ a Λ a (x) ψ(x) τ a : Pauli matrices µ ψ = e i 2 τ a Λ a ( i(τ a µ Λ a ) + µ ) ψ define D µ µ i gτ a A a (x) 2 g: coupling constant, A a (x): gauge field Fµν a = [ ] 1 g Dµ, D ν = µ A ν ν A µ + ɛ abc A b µ Ac ν last term because SU(2) non-abelian December 16, 2010 Institut für Kernphysik, TU Darmstadt Pascal Büscher 7

6 Spontaneous Symmetry Breaking Scalar Potential V (ϕ) = 1 2 m2 ϕ λ2 ϕ 4 minimum at ϕ = ± m λ = ±v V (ϕ) = 1 4 λ2 ( ϕ 2 v 2) 2 change of ϕ along valley w/o change of V Goldstone boson December 16, 2010 Institut für Kernphysik, TU Darmstadt Pascal Büscher 8 (

7 Higgs Mechanism in U(1) ( L = D µ ϕ λ2 ϕ 2 v ) F µνf µν choose unitary gauge: ϕ(x) = 1 2 (v + χ(x)) real everywhere L = 1 2 ( µ χ(x) ) 2 + e 2 gauge boson A µ gains mass ev 2 A µa µ ( v 2 + 2vχ(x) + χ 2 (x) ) λ2 8 (2v + χ(x))2 χ 2 (x) 1 4 F µνf µν December 16, 2010 Institut für Kernphysik, TU Darmstadt Pascal Büscher 9

8 Higgs Mechanism in SU(2) ( L = D µ ϕ λ2 ϕ 2 v ) F µν a F aµν choose unitary gauge with ϕ(x) = 1 2 ( from kinetic term: g 2 8 τ a Wµ a ( 0 v with W µ ± = ( ) 1 2 W 1 µ iwµ 2 all gauge bosons gain M W = gv 2 ) 2 ( gv = 2 0 v + h(x) ) ) 2 [ W + µ W µ W 3 µ W 3µ] December 16, 2010 Institut für Kernphysik, TU Darmstadt Pascal Büscher 10

9 Glashow-Weinberg-Salam (GWS) Theory Requirements: beta decay: off-diagonal generator to describe e.g. u going to d simplest case: SU(2) photon does not couple to neutrino cannot be connected τ 3 of SU(2) Next simplest construction: SU(2) U(1) December 16, 2010 Institut für Kernphysik, TU Darmstadt Pascal Büscher 12

10 Electroweak Higgs Mechanism U(1) SU(2) Covariant derivative: D µ = µ i g τ a W a 2 µ i g 2 YB µ D µ ϕ 2 contains: ( gv ) 2 W + 2 µ W µ + v ( ) 2 W 3 T ( µ g 2 gg 8 gg g 2 B µ ) ( W 3 µ B µ ) W ± gain mass M W ± = gv 2 diagonalize matrix A µ = cos θ W B µ + sin θ W W 3 µ, M A = 0 Z µ = sin θ W B µ + cos θ W W 3 µ, M Z = v 2 with cos θ W = g = MW g+g M Z by comparison with experiment sin 2 θ W 0.23, v = 246 GeV December 16, 2010 Institut für Kernphysik, TU Darmstadt Pascal Büscher 13 g 2 + g 2

11 Mass Generation of Fermions ( ) ( ψ1 u Fermions: left-handed SU(2)-doublets ψ L =, e.g. Q ψ L = 2 d L right-handed SU(2)-singlet ψ 1R, ψ 2R, e.g. u R, d R with Yukawa interaction e.g. for quarks: L Y = Γ ij d Q i L ϕd j R + h.c. Γ ij u Q i L ϕ c u j R + h.c. : lin. comb. of quark mass eigenstates i, j: indices of generation space Mass matrix: M ij v 2 Γ ij diagonalize M ij L Y = ) m f ( f L f R + f R f L f ) L December 16, 2010 Institut für Kernphysik, TU Darmstadt Pascal Büscher 14

12 Higgs Mass Mass term for the Higgs boson in potential V ( ) With ϕ(x) = and V (ϕ) = ( 1 v + h(x) 8 λ2 h 2 2hv ) 2 mass term: 1 2 λ2 v 2 h 2 Higgs boson has (bare) mass of m h = λv λ only in V (ϕ) m h not deducible from so far observed quantities However: Theory provides us with constraints December 16, 2010 Institut für Kernphysik, TU Darmstadt Pascal Büscher 15

13 Renormalization-Group Flow Parameters of theory (e.g. coupling constants) set at a certain energy scale Q change energy scale include dressing functions into paramters e.g.: parameters flow as the energy scale is changed December 16, 2010 Institut für Kernphysik, TU Darmstadt Pascal Büscher 16

14 Constraints on the Higgs mass from Theory Triviality With λ 1 2 λ2 the coupling evolves with (only scalar term) dλ d ln(q 2 ) = 3λ 2 4π 2 Q: energy scale λ (Q) = 1 3λ (Q 0) 4π 2 λ (Q 0 ) ( ln Q 2 Q 2 0 ) pole at Q c = Q 0 e 4π2 3λ (Q 0 ) SM perturbative as long as pole Q c > Λ λ (v) 4π 2 m 2 h < 8π 2 v 2 3 ln ( Λ 2 /v 2) 3 ln(λ 2 /v 2 ) for Λ = GeV: m h < 160 GeV including effects from coupling: m h < 170 GeV December 16, 2010 Institut für Kernphysik, TU Darmstadt Pascal Büscher 17

15 Constraints on the Higgs mass from Theory Vacuum Stability V (ϕ) needs lower bound λ > 0 for any scale RG for small λ : dλ [ d ln(q 2 ) = 1 48 m4 t 16π 2 v + 3 ( ) ] 2g 4 + (g 2 + g 2 ) Solving RG up to two-loop level: m h > 134 GeV for Λ = GeV December 16, 2010 Institut für Kernphysik, TU Darmstadt Pascal Büscher 18

16 Constraints on the Higgs mass from Theory ( K. Riesselmann, [hep-ph/ ].) December 16, 2010 Institut für Kernphysik, TU Darmstadt Pascal Büscher 19

17 Problems So, is everything settled then? Higgs not found yet Problems with quadratic divergences in mass corrections December 16, 2010 Institut für Kernphysik, TU Darmstadt Pascal Büscher 20

18 Quadratic Divergences Divergent corrections to m h : h f h ( 2h 2 f ) Λ2 16π 2 with Yukawa coupling h f h h h λ Λ2 16π 2 λ = 2hf 2 to cancel on one-loop level Problem: not for higher loops µ 2 µ = µ λλ 2 absorb in bare mass term (quadratic term in V ) Problem: to obtain m h 100 GeV at LHC energies very large bare mass necessary unnatural fine-tuning December 16, 2010 Institut für Kernphysik, TU Darmstadt Pascal Büscher 21

19 Possible Solutions Various ansätze: Technicolour Supersymmetric ansätze Higgsless scenarios... December 16, 2010 Institut für Kernphysik, TU Darmstadt Pascal Büscher 23

20 Technicolour Idea Motivation consider self-energy from pions: e.g. W ± π ± W ± Π(q 2 ) g2 2 f 2 π for small q 4q 2 in propagator: 1 q 1 2 q 2 (1 + Π(q 2 )) = 1 pion contribution leads to boson mass of 1 4 g 2f π q 2 + g2 2 f 2 π 4 Boson mass orders of magnitude too small with physical pions introduce technipions and technifermions, f π F π December 16, 2010 Institut für Kernphysik, TU Darmstadt Pascal Büscher 24

21 Technicolour Features F π takes role of v technipion composite particle substructure visible above a few TeV dynamics of fermions involved December 16, 2010 Institut für Kernphysik, TU Darmstadt Pascal Büscher 25

22 MSSM - A Supersymmetric Scenario symmetry transformation with a spinor as generator such that bosons fermions superpartner cancel quadratic divergences: sign opposite (fermionic/bosonic) identical mass ((unbroken) Supersymmetry) two Higgs doublets with opposite hypercharge 5 higgs bosons: 2 CP even Higgs h, H, 2 charged Higgs H ± 1 CP odd Higgs A m h = 140 GeV..200 GeV reasonable RG flow of µ Huge Problem superpartners not measured supersymmetry broken December 16, 2010 Institut für Kernphysik, TU Darmstadt Pascal Büscher 26

23 A Higgless (Toy) Scenario Assume 5d-spacetime 5th dim. compactified on circle with symmetry y y fixed points at y = 0, πr boundary condition for a 3-component boson field 5A a µ y=0 = 0 A 1,2 µ = 0, y=πr 5A3 µ y=πr = 0 Solution: Kaluza-Klein mode gauge fields: A 3 µ(y) cos( ny ) R Mn = 0, 1, 2,... R R A 1,2 µ (y) cos( (2m+1)y ) M 2R m = 1, 3,... 2R 2R massive bosons without extra particles more realistic masses with more sophisticated boundary conditions R y December 16, 2010 Institut für Kernphysik, TU Darmstadt Pascal Büscher 27

24 Summary Higgs mechanism explains massive gauge bosons mass generation of fermions within the Standard model (SM) Higgs mass cannot be extracted directly from SM, but constraints can Quadratic divergences pose (aesthetic) problem Alternatives (more or less) able to explain mass generation Different concepts with no/many Higgs bosons (non-)measurement of the Higgs boson at the LHC should provide new insights on how electroweak physics really works December 16, 2010 Institut für Kernphysik, TU Darmstadt Pascal Büscher 29

25 Thanks for your attention! Merry Christmas & a Happy New Year! December 16, 2010 Institut für Kernphysik, TU Darmstadt Pascal Büscher 30

26 flavor mixing via charged currents Flavor mixing from charged current interaction: transform into mass eigensystem e 2 sin θw ū i L /W + d i L + h.c. e ū i L V CKM /W + d i L + h.c. 2 sin θw Flavor changing due to transformation matrix V CKM Cabibbo-Kobayashi-Maskawa matrix December 16, 2010 Institut für Kernphysik, TU Darmstadt Pascal Büscher 32