To Higgs or not to Higgs

Transcription

1 To Higgs or not to Higgs vacuum stability and the origin of mass Wolfgang Gregor Hollik DESY Hamburg Theory Group Dec MU Programmtag Mainz

2 The Higgs mechanism and the origin of mass [CERN bulletin]

3 The Higgs mechanism and the origin of mass Massive Gauge Bosons The tale of the Higgs dale potential instable at the origin ground state (= vacuum) breaks symmetry spontaneously broken SU(2) L U(1) Y ( ) ( ) φ + 0 Φ(x) = v h(x) φ 0 (D µ φ) D µ φ quadratic terms for gauge fields: 1 4 v2 ( ga 3 µ g B µ ) ( ga 3 µ g B µ) g2 v 2 A + µ A µ, with A ± µ = 1 2 (A 1 µ ± ia 2 µ).

4 We can do more! Masses for W and Z weak mixing angle m 2 W = v2 g 2 Fundamental masses 2, m 2 Z = v2 2 ( g 2 + g 2 ), tan θ w = g g, m W = cos θ w m Z. all fundamental masses v = 246 GeV in the Standard Model Masses for fermions Yukawa interactions: L Yuk = Y Ψ L ΦΨ R + h. c. = Y ( ) ( ) 0 ψu L ψd L v/ ψr d + h. c. 2 (for up-type fields: Φ Φ = iσ 2 Φ and ψ d R ψu R ) m ψ = v 2 Y

5 Origin of mass = origin of v In the Standard Model, by construction, ( ) 2 V SM = µ 2 H H + λ H H leads to v 2 = µ 2 /λ (with µ 2 > 0, λ > 0). So what? measuring gauge boson masses + gauge couplings: v 2 measuring Higgs boson mass: λ only tree level no new physics? V SM put by hand... going beyond tree and SM: minimum may get unstable new minimum: different v = different mass may break additional good symmetries of the SM

6 Vacuum Stability: Constraining inconsistency The Standard Model (In)Stability V SM = µ 2 H H + λ large field values: V λ(h H) 2 RGE: λ λ(q), where Q H ( ) 2 H H λ 0 around Q GeV, new minimum beyond M Planck Connection to Matter & Universe transplanckian vev: different physics particle masses beyond M Planck : link to gravitational physics Higgs inflation? [Bezrukov, Rubio, Shaposhnikov 14] main sources for uncertainty: m t and α S

7 Precise analysis: up to three loops! [Zoller 2014]

8 Precise analysis: up to three loops! [Zoller 2014]

9 Stability, instability or metastability? [Courtesy of Max Zoller]

10 Stability, instability or metastability? [Courtesy of Max Zoller]

11 The SM phase diagram 200 Instability Top mass Mt in GeV Meta stability Stability Non perturbativity [Degrassi et al. JHEP 1208 (2012) 098] h

12 Vacuum stability beyond the Standard Model Quantum effects: new particles in the loop tree level: multi-scalar potentials Two Higgs Doublet Models (2HDM): no charge breaking 2HDM + Singlet: more involved Minimal Supersymmetric Standard Model (MSSM): sfermions: additional electrically and color charged directions NMSSM: additional non-trivial neutral vacua severe constraints for any model, can be tested numerically generically difficult to get practical limits i.e. noanalytical bounds if so, with many simplifications (see next slides) [cf. Vevacious]

13 Vacuum stability beyond the Standard Model Quantum effects: new particles in the loop tree level: multi-scalar potentials Two Higgs Doublet Models (2HDM): no charge breaking 2HDM + Singlet: more involved Minimal Supersymmetric Standard Model (MSSM): sfermions: additional electrically and color charged directions NMSSM: additional non-trivial neutral vacua severe constraints for any model, can be tested numerically generically difficult to get practical limits i.e. noanalytical bounds if so, with many simplifications (see next slides) [cf. Vevacious]

14 The MSSM tree-level scalar potential, 3rd generation squarks V q,h = t L ( m 2 L + y t h 2 2) t L + t R ( m 2 t + y t h 2 2) t R + b L ( m 2 L + y b h 1 2) bl + b R ( m 2 b + y bh 1 2) br [ t L (µ y t h 1 A t h 2 ) t R + h.c. ] [ bl (µ y b h 2 A b h 1 ) b R + h.c. ] + y t 2 t L 2 t R 2 + y b 2 b L 2 b R 2 + y t 2 b L 2 t R 2 + y b 2 t L 2 b R 2 + g2 1 ( h 2 2 h b L b R t L 2 43 ) 2 t R 2 + g g2 3 ( h 2 2 h b L 2 t L 2) 2 + g 2 2 ( t L 2 t R 2 + b L 2 b R 2) 2 2 t L 2 b L (m 2 h 2 + µ 2 ) h 2 2 +(m 2 h 1 + µ 2 ) h Re(B µ h 1 h 2 ).

15 The MSSM tree-level scalar potential, 3rd generation squarks V q,h = t L ( m 2 L + y t h 2 2) t L + t R ( m 2 t + y t h 2 2) t R + b L ( m 2 L + y b h 1 2) bl + b R ( m 2 b + y bh 1 2) br [ t L (µ y t h 1 A t h 2 ) t R + h.c. ] [ bl (µ y b h 2 A b h 1 ) b R + h.c. ] + y t 2 t L 2 t R 2 + y b 2 b L 2 b R 2 + y t 2 b L 2 t R 2 + y b 2 t L 2 b R 2 + g2 1 ( h 2 2 h b L b R t L 2 43 ) 2 t R 2 + g g2 3 ( h 2 2 h b L 2 t L 2) 2 + g 2 2 ( t L 2 t R 2 + b L 2 b R 2) 2 2 t L 2 b L (m 2 h 2 + µ 2 ) h 2 2 +(m 2 h 1 + µ 2 ) h Re(B µ h 1 h 2 ). t L = t R = t, b L = b R = b

16 The MSSM tree-level scalar potential, 3rd generation squarks V q,h = t ( m 2 L + y t h 2 2) t + t ( m 2 t + y t h 2 2) t + b ( m 2 L + y b h 1 2) b + b ( m 2 b + y bh 1 2) b [ t (µ y t h 1 A t h 2 ) t + h.c. ] [ b (µ y b h 2 A b h 1 ) b + h.c. ] + y t 2 t 2 t 2 + y b 2 b 2 b 2 + y t 2 b 2 t 2 + y b 2 t 2 b 2 ) 2 + g2 1 ( h 2 2 h b 2 t g2 2 8 ( h 2 2 h b 2 t 2) 2 + g t 2 b 2 + (m 2 h 2 + µ 2 ) h 2 2 +(m 2 h 1 + µ 2 ) h Re(B µ h 1 h 2 ). t L = t R = t, b L = b R = b

17 The MSSM tree-level scalar potential, 3rd generation squarks V q,h = t ( m 2 L + y t h 2 2) t + t ( m 2 t + y t h 2 2) t + b ( m 2 L + y b h 1 2) b + b ( m 2 b + y bh 1 2) b [ t (µ y t h 1 A t h 2 ) t + h.c. ] [ b (µ y b h 2 A b h 1 ) b + h.c. ] + y t 2 t 2 t 2 + y b 2 b 2 b 2 + y t 2 b 2 t 2 + y b 2 t 2 b 2 ) 2 + g2 1 ( h 2 2 h b 2 t g2 2 8 ( h 2 2 h b 2 t 2) 2 + g t 2 b 2 + (m 2 h 2 + µ 2 ) h 2 2 +(m 2 h 1 + µ 2 ) h Re(B µ h 1 h 2 ). t L = t R = t, b L = b R = b ; b = h 1 = φ 1, t = h 2 = φ 2

18 The MSSM tree-level scalar potential, 3rd generation squarks V q,h = φ 2 ( m 2 L + y t φ 2 2) φ 2 + φ 2 ( m 2 t + y t φ 2 2) φ 2 +φ 1 ( m 2 L + y b φ 1 2) φ 1 + φ 1 ( m 2 b + y bφ 1 2) φ 1 [ φ 2 (µ y t φ 1 A t φ 2 ) φ 2 + h.c. ] [ φ 1 (µ y b φ 2 A b φ 1 ) φ 1 + h.c. ] + y t 2 φ 2 2 φ y b 2 φ 1 2 φ y t 2 b 2 t 2 + y b 2 t 2 b 2 + g2 2 2 t 2 b 2 + (m 2 h 2 + µ 2 ) φ 2 2 +(m 2 h 1 + µ 2 ) φ Re(B µ φ 1 φ 2 ). t L = t R = t, b L = b R = b ; b = h 1 = φ 1, t = h 2 = φ 2

19 The MSSM tree-level scalar potential, 3rd generation squarks V q,h = φ 2 ( m 2 L + y t φ 2 2) φ 2 + φ 2 ( m 2 t + y t φ 2 2) φ 2 [ φ 2 ( A t φ 2 ) φ 2 + h.c. ] + y t 2 φ 2 2 φ y t 2 b 2 t 2 + y b 2 t 2 b 2 g t 2 b 2 + (m 2 h 2 + µ 2 ) φ 2 2. t L = t R = t, b L = b R = b ; b = h 1 = φ 1, t = h 2 = φ 2

20 Mathematics for the kindergarden Minimize the potential V (φ) = m 2 φ 2 Aφ 3 + λφ 4, with m 2 = m 2 h 2 + µ 2 + m 2 L + m2 t, A = A t and λ = 3y 2 t.

21 Mathematics for the kindergarden Minimize the potential V (φ) = m 2 φ 2 Aφ 3 + λφ 4, with m 2 = m 2 h 2 + µ 2 + m 2 L + m2 t, A = A t and λ = 3y 2 t. Answer: φ 0 = 0, φ ± = 3A ± 9A 2 32λ m 2. 8λ Condition to be safe from non-standard (i.e. non-trivial) minima: V (φ ± ) > 0 m 2 > A2 4λ

22 Mathematics for the kindergarden Minimize the potential V (φ) = m 2 φ 2 Aφ 3 + λφ 4, with m 2 = m 2 h 2 + µ 2 + m 2 L + m2 t, A = A t and λ = 3y 2 t. Answer: φ 0 = 0, φ ± = 3A ± 9A 2 32λ m 2. 8λ Condition to be safe from non-standard (i.e. non-trivial) minima: V (φ ± ) > 0 m 2 > A2 4λ Well-known constraints [Gunion, Haber, Sher 88] A t 2 < 3yt 2 ( m 2 h2 + µ 2 + m 2 L + m 2 ) t A b 2 < 3yb 2 ( m 2 h1 + µ 2 + m 2 L + m 2 b) for the limiting cases t L = t R = h 2 and b L = b R = h 1!

23 A 2 < 4λm 2

24 A 2 = 4λm 2

25 A 2 > 4λm 2

26 A simple view of a complicated object h 2 = φ, t = α φ, h 1 = ηφ, b = β φ V φ = ( m 2 h 2 + η 2 m 2 h 1 + (1 + η 2 )µ 2 2B µ η + (α 2 + β 2 ) m 2 L + α 2 m 2 t + β 2 m 2 ) b φ 2 2 ( α 2 (µy t η A t ) + β 2 (µy t ηa b ) ) φ 3 + (α 2 yt 2 + β 4 yb 2 ( )φ4 g g2 2 ) (1 η 2 + β 2 α 2 ) 2 + 2α 2 yt 2 + 2β 2 yb 2 φ 4 8 M 2 (η, α, β)φ 2 A(η, α, β)φ 3 + λ(η, α, β)φ 4,

27 A simple view of a complicated object h 2 = φ, t = α φ, h 1 = ηφ, b = β φ V φ = ( m 2 h 2 + η 2 m 2 h 1 + (1 + η 2 )µ 2 2B µ η + (α 2 + β 2 ) m 2 L + α 2 m 2 t + β 2 m 2 ) b φ 2 with 2 ( α 2 (µy t η A t ) + β 2 (µy t ηa b ) ) φ 3 + (α 2 yt 2 + β 4 yb 2 ( )φ4 g g2 2 ) (1 η 2 + β 2 α 2 ) 2 + 2α 2 yt 2 + 2β 2 yb 2 φ 4 8 M 2 (η, α, β)φ 2 A(η, α, β)φ 3 + λ(η, α, β)φ 4, M 2 = m 2 h 2 + η 2 m 2 h 1 + (1 + η 2 )µ 2 2B µ η + (α 2 + β 2 ) m 2 L + α 2 m 2 t + β 2 m 2 b, A = 2α 2 ηµy t 2α 2 A t + 2β 2 µy b 2ηβ 2 A b, λ = g2 1 + g2 2 (1 η 2 + β 2 α 2 ) (2 + α 2 )α 2 yt 2 + (2η 2 + β 2 )β 2 yb 2.

28 Optimized Charge and Color Breaking [Gunion, Haber, Sher 88; Casas, Lleyda, Muñoz 96] The same but different ( A-parameter bounds ) A 2 < 4λM 2 4λ(η, α, β)m 2 (η, α, β) > (A(η, α, β)) 2 h u = b, h 0 d = 0 m 2 H u + µ 2 + m 2 Q + m 2 b > (µy b ) 2 y 2 b + (g2 1 + g2 2 )/2 [WGH 15] h d 2 = h u 2 + b 2, b = αh u [WGH 15] m 2 11(1 + α 2 ) + m 2 22 ± 2m α 2 + α 2 ( m 2 Q + m 2 b ) > 4µ2 α α 2

29 Closing in on the parameter space A b = 0 GeV

30 Closing in on the parameter space A b = 500 GeV

31 Closing in on the parameter space A b = 1000 GeV

32 Closing in on the parameter space A b = 1500 GeV

33 The issue of including field directions t = 0, h d = 0, A b = 0

34 The issue of including field directions t = 0, A b = 0

35 The issue of including field directions A b = 0

36 The issue of including field directions A b = A t

37 Resort comes close (increasing MSUSY)

38 Resort comes close (increasing MSUSY) A b = 0

39 Parameter choice crucial µ = 350 GeV

40 Parameter choice crucial µ = M MSUSY

41 Parameter choice crucial µ = 500 GeV

42 Parameter choice crucial µ = 500 GeV

43 Short summary Vacuum stability and the origin of mass: v = 246 GeV vs. v > M Planck or ṽ M SUSY constraints on model parameters from theoretical consistency: global minimum has to be electroweak minimum heavy light 125 GeV: large SUSY corrections e.g. MSSM: large A t,b and µ induce squark vevs

44 Backup Slides

45 A comment on metastability and quantum tunneling Cosmological stability bounce action B 400 life-time longer than age of the universe Decay probability (per unit volume) Γ V = Ae B/ [Coleman 77] Death and doom value of B crucially depends on field space path very different conclusions for different η, α, β independent of SUSY parameter choice

46 Contours of the bounce µ = 500 GeV, A b = A t = 1500 GeV

47 Contours of the bounce µ = 500 GeV, A b = A t = 1500 GeV

48 Contours of the bounce µ = A b = A t = 500 GeV

49 Contours of the bounce µ = A b = A t = 500 GeV