To Higgs or not to Higgs - PDF Free Download

To Higgs or not to Higgs vacuum stability and the origin of mass Wolfgang Gregor Hollik DESY Hamburg Theory Group Dec 12 2016 MU Programmtag Mainz

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

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 + 1 2 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) + 1 2 g2 v 2 A + µ A µ, with A ± µ = 1 2 (A 1 µ ± ia 2 µ).

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

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

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 10 10 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

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

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

The SM phase diagram 200 Instability Top mass Mt in GeV 150 100 50 0 Meta stability Stability Non perturbativity 0 50 100 150 200 [Degrassi et al. JHEP 1208 (2012) 098] h

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]

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 1 2 + 13 8 b L 2 + 23 b R 2 + 13 t L 2 43 ) 2 t R 2 + g2 2 8 + g2 3 ( h 2 2 h 1 2 + 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 2 8 + (m 2 h 2 + µ 2 ) h 2 2 +(m 2 h 1 + µ 2 ) h 1 2 2 Re(B µ h 1 h 2 ).

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 1 2 + b 2 t 2 8 + g2 2 8 ( h 2 2 h 1 2 + b 2 t 2) 2 + g 2 2 2 t 2 b 2 + (m 2 h 2 + µ 2 ) h 2 2 +(m 2 h 1 + µ 2 ) h 1 2 2 Re(B µ h 1 h 2 ). t L = t R = t, b L = b R = b

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 2 2 + y b 2 φ 1 2 φ 1 2 + 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 1 2 2 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

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 2 2 + 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. t L = t R = t, b L = b R = b ; b = h 1 = φ 1, t = h 2 = φ 2

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.

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!

A 2 < 4λm 2

A 2 = 4λm 2

A 2 > 4λm 2

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 2 + 1 + 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,

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 2 + 1 + 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 8 + (2 + α 2 )α 2 yt 2 + (2η 2 + β 2 )β 2 yb 2.

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 2 12 1 + α 2 + α 2 ( m 2 Q + m 2 b ) > 4µ2 α 2 2 + 3α 2

Closing in on the parameter space A b = 0 GeV

Closing in on the parameter space A b = 500 GeV

Closing in on the parameter space A b = 1000 GeV

Closing in on the parameter space A b = 1500 GeV

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

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

The issue of including field directions A b = 0

The issue of including field directions A b = A t

Resort comes close (increasing MSUSY)

Resort comes close (increasing MSUSY) A b = 0

Parameter choice crucial µ = 350 GeV

Parameter choice crucial µ = M MSUSY

Parameter choice crucial µ = 500 GeV

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 Higgs @ 125 GeV: large SUSY corrections e.g. MSSM: large A t,b and µ induce squark vevs

Backup Slides

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

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

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