Dark Matter and Neutrino Masses from a Scale-Invariant Multi-Higgs Portal Alexandros Karam Theory Division Physics Department University of Ioannina AK and Kyriakos Tamvakis: 1508.03031 September 7, 015 Alexandros Karam (Univeristy of Ioannina) 1508.03031 September 7, 015 1 / 8
Beyond the Standard Model (SM) Shortcomings of the SM: Dark Matter Neutrino Masses Vacuum Stability Matter-Antimatter Asymmetry Flavor Problem Strong CP Problem Gauge Unification Inflation... Alexandros Karam (Univeristy of Ioannina) 1508.03031 September 7, 015 3 / 8
The Hierarchy Problem Solutions to the problems of the SM generally require new large scales Λ NP υ EW. These scales contain particles with mass m = Λ NP that give loop corrections to the Higgs boson mass: M h = M 0 + m h, m h Λ NP A huge amount of fine-tuning is needed between M0 and Λ NP in order to obtain M h = 15.09 GeV. A similar argument holds if we consider the SM as an effective theory, without any additional degrees of freedom, having a cut-off Λ UV. Alexandros Karam (Univeristy of Ioannina) 1508.03031 September 7, 015 4 / 8
The Hierarchy Problem Popular Solution: Supersymmetry Alternate Solution: Classical Scale Symmetry Alexandros Karam (Univeristy of Ioannina) 1508.03031 September 7, 015 5 / 8
Classical Scale Invariance (CSI) A theory is said to have Classical Scale Symmetry if the Lagrangian is invariant under the transformations x µ dx µ, S(x) ds(dx), V µ (x) dv µ (dx), F (x) d 3/ F (dx) There is only one scale in the SM, the parameter µ SM in the scalar potential V (H) = 1 µ SMH H + λ(h H) With µ SM = 0 the SM is Classically Scale Invariant (T µ µ ) classical = 0. The quadratic sensitivity to the cut-off is now unphysical. Quantum effects break the symmetry through the β-functions (T µ µ = β λi O i ), but this does not reintroduce the quadratic divergences. = Solution to the Hierarchy Problem Alexandros Karam (Univeristy of Ioannina) 1508.03031 September 7, 015 6 / 8
Coleman-Weinberg Mechanism (1973) Massless scalar QED: L = 1 4 F µνf µν + (D µ φ) D µ φ λ φ 4! φ 4 1-loop effective potential: [ V eff (φ) = λ ( ) ] φ 4 φ 4 + 3e4 φ φ 64π φ 4 log φ 5 6 Minimization at φ = φ 0 gives λ φ ( φ ) = 33 8π e4 φ ( φ ) ( ) φ V eff (φ) e 4 φ [log φ 4 φ 1 ] Alexandros Karam (Univeristy of Ioannina) 1508.03031 September 7, 015 7 / 8
Coleman-Weinberg Mechanism (1973) Dimensional Transmutation: exchange a dimensionless parameter λ φ for a dimensionful parameter φ. [ ] λ φ ( φ ) = 33 8π e4 φ ( φ ), The scalar boson (scalon) obtains a mass: φ Λ UV exp 4π e φ ( φ ) Λ UV M s = 3e4 φ 8π φ In the SM: M s 1 [ 6M 4 8π υ W + 3MZ 4 1M 4 ] t < 0 EW Alexandros Karam (Univeristy of Ioannina) 1508.03031 September 7, 015 8 / 8
Conclusion We should add bosonic fields in order to obtain a positive mass. Alexandros Karam (Univeristy of Ioannina) 1508.03031 September 7, 015 9 / 8
Gildener-Weinberg Formalism (1976) V 0 (Φ) = 1 4 λ ijklφ i Φ j Φ k Φ l We choose a renormalization scale Λ, at which V 0 (Φ) has a nontrivial minimum on some ray Φ i = N i ϕ: min V 0(N) = min (λ ijkl(λ)n i N j N k N l ) = 0 N i N i =1 N i N i =1 The conditions for a minimum along a particular direction N i = n i are V 0 N i = V 0 (n) = 0 n 1-loop potential V 1 (nϕ) = Aϕ 4 + Bϕ 4 log ϕ Λ. Alexandros Karam (Univeristy of Ioannina) 1508.03031 September 7, 015 10 / 8
Gildener-Weinberg Formalism (1976) A = { [ 1 64π ϕ 4 3 Tr MV 4 ln M V ] [ ϕ +Tr MS 4 ln M S ] [ ϕ 4 Tr MF 4 ln M F ] } ϕ B = 1 ( 3 Tr M 4 64π ϕ 4 V + Tr MS 4 4 Tr MF 4 ) The symmetry breaks when: V 1 (nϕ) ϕ = 0 ln ϕ ϕ= ϕ Λ = 1 A B Quantum corrections along the flat direction give mass to the scalar: M s = 1 ( 3 Tr M 4 8π ϕ V + Tr MS 4 4 Tr MF 4 ) Alexandros Karam (Univeristy of Ioannina) 1508.03031 September 7, 015 11 / 8
The Model CSI SU(3) c SU() L U(1) Y SU() X New fields: 1 complex scalar doublet Φ under SU() X 3 gauge bosons X a 1 real scalar singlet σ 3 right-handed neutrinos N i Unitary gauge: H = 1 ( 0 h ), Φ = 1 ( 0 φ ) V 0 (h, φ, σ) = λ h 4 h4 + λ φ 4 φ4 + λ σ 4 σ4 λ hφ 4 h φ λ φσ 4 φ σ + λ hσ 4 h σ L N = Yν ij L i iσ H N j + h.c. + Yσ ij N i c N j σ Alexandros Karam (Univeristy of Ioannina) 1508.03031 September 7, 015 1 / 8
Vacuum Stability The scalar potential is bounded from below if the matrix λ h λ hφ λ hσ A = 1 8 λ hφ λ φ λ φσ λ hσ λ φσ λ σ is copositive, i.e. such that η a A ab η b is positive for non-negative vectors in the basis (h, φ, σ ). This is equivalent to the stability conditions λ hφ λ h λ φ 1, det A = λ h λ φ λ σ 1 4 λ h 0, λ φ 0, λ σ 0 λ hσ λ h λ σ 1, λ φσ λ φ λ σ 1 ( λ hφ λ σ + λ hσ λ φ + λ φσ λ h) + 1 4 λ hφλ hσ λ hσ 0 Alexandros Karam (Univeristy of Ioannina) 1508.03031 September 7, 015 13 / 8
Flat Direction In order to study the flat directions of the tree-level potential we may parametrize the scalar fields as h = ϕn 1, φ = ϕn, σ = ϕn 3, with N i a unit vector in the three-dimensional field space. The condition for an extremum along a particular direction N i = n i is V 0 N i = V 0 (n) = 0 n Then, the equations giving the symmetry breaking direction are λ h n 1 = λ hφ n λ hσ n 3 λ φ n = λ hφ n 1 + λ φσ n 3 λ σ n 3 = λ φσ n λ hσ n 1 λ h n 4 1 + λ φ n 4 + λ σ n 4 3 λ hφ n 1n λ φσ n n 3 + λ hσ n 1n 3 = 0 Alexandros Karam (Univeristy of Ioannina) 1508.03031 September 7, 015 14 / 8
The shifted scalar fields and vevs are Mass Matrix h = (ϕ + v) n 1, φ = (ϕ + v) n, σ = (ϕ + v) n 3 h v h = v n 1, φ v φ = v n, σ v σ = v n 3 From the shifted tree-level potential we can read off the scalar mass matrix λ h n M 0 = υ 1 n 1 n λ hφ +n 1 n 3 λ hσ n 1 n λ hφ λ φ n n n 3 λ φσ +n 1 n 3 λ hσ n n 3 λ φσ λ σ n 3 in the (h, φ, σ) basis. Introduce a general rotation in terms of three parametric angles R M 0 R 1 = M d, with the rotation matrix R 1 given by ( cos α cos β sin α cos α sin β ) R 1 = cos β cos γ sin α + sin β sin γ cos α cos γ cos γ sin α sin β cos β sin γ cos γ sin β cos β sin α sin γ cos α sin γ cos β cos γ sin α sin β sin γ Alexandros Karam (Univeristy of Ioannina) 1508.03031 September 7, 015 15 / 8
Parametrize the vevs as ( h1 h h 3 Mass Eigenstates ) = Then, M d is diagonal, provided that (.... R.... ) ( h φ σ v h = v sin α = vn 1 v φ = v cos α cos γ = vn v σ = v cos α sin γ = vn 3 ) v h tan α = vφ + v σ = 4λ φ λ σ λ φσ (λ σλ hφ λ φ λ hσ ) + λ φσ (λ hφ λ hσ ) tan γ = v σ = λ hλ φσ λ hφ λ hσ vφ 4λ h λ σ λ hσ tan β = v h v φ v σv (λ hσ + λ hφ ) (λ φ + λ σ + λ φσ ) v φ v σ λ h v h v. Alexandros Karam (Univeristy of Ioannina) 1508.03031 September 7, 015 16 / 8
Mass Eigenvalues The resulting mass eigenvalues are Mh 1 / = λ h vh cos α cos β + λ φ vφ (cos β cos γ sin α sin β sin γ) +λ σvσ (cos γ sin β + cos β sin α sin γ) +λ hφ v h v φ cos α cos β (cos β cos γ sin α sin β sin γ) λ φσ v φ v σ (cos β cos γ sin α sin β sin γ) (cos γ sin β + cos β sin α sin γ) λ hσ v h v σ cos α cos β (cos γ sin β + cos β sin α sin γ) Mh = 0 Mh 3 / = λ h vh cos α sin β + λ φ vφ (sin β cos γ sin α + cos β sin γ) +λ σvσ (cos γ cos β sin β sin α sin γ) +λ hφ v h v φ cos α sin β (sin β cos γ sin α + cos β sin γ) +λ φσ v φ v σ (sin β cos γ sin α + cos β sin γ) (cos γ cos β sin β sin α sin γ) +λ hσ v h v σ cos α sin β (cos γ cos β sin β sin α sin γ) Alexandros Karam (Univeristy of Ioannina) 1508.03031 September 7, 015 17 / 8
Neutrino Masses The Yukawa terms that give rise to neutrino masses are Y ij ν v h ν i iσ N j + h.c. + Y ij σ v σ N c i N j The neutrino mass matrix 0 Y ν v h Y ν v h Y σv σ has the following eigenvalues M N Y σ v σ, m ν v ( ) h Y ν Y 1 σ Yν, 4v σ with M N O(100 GeV) and m ν O(0.1 ev), assuming Y ν v h O(10 4 GeV), v σ O(1 TeV) and Y σ O(0.1), so that Y ν v h Y σ v σ. Alexandros Karam (Univeristy of Ioannina) 1508.03031 September 7, 015 18 / 8
The 1-loop Potential Along the minimum flat direction at the scale Λ: where A = 1 ( 64π υ 4 Mh 4 1,3 3 + log M i υ 1,3 ( ) V 1 (nϕ) = Aϕ 4 + Bϕ 4 log ϕ Λ, ) ( + 6MW 4 5 6 + log M ) ( W υ + 3MZ 4 5 6 + log M ) Z υ +9MX 4 5 6 + log M X ( υ 1Mt 4 1 + log M t ) υ 6MN 4 1 + log M N υ 1 B = 64π υ 4 Mh 4 1,3 + 6MW 4 + 3M Z 4 + 9M X 4 1M t 4 6MN 4 1,3 Minimization: V 1 (nϕ) ϕ ( υ ) = 0 log ϕ=v Λ ( = 1 4 A B )], Alexandros Karam (Univeristy of Ioannina) 1508.03031 September 7, 015 19 / 8
Darkon Mass The one-loop effective potential becomes [ V 1 (nϕ) = Bϕ 4 log ϕ υ 1 ] The pseudo-goldstone boson (darkon) mass is now shifted from zero to Mh = 1 ( M 4 8π υ h1 + Mh 4 3 + 6MW 4 + 3MZ 4 + 9MX 4 1Mt 4 6MN 4 ), Positivity condition B > 0 translates to v = v h + v φ + v σ M 4 h 3 + 9M 4 X 6M 4 N > (317.6 GeV) 4 Alexandros Karam (Univeristy of Ioannina) 1508.03031 September 7, 015 0 / 8
Model RGEs β g1 = 41 10 g3 1 + 1 1 ( (4π) 50 g3 1 199g ) 1 + 135g + 440g 3 85y t β g = 19 6 g3 + 1 (4π) 1 30 g3 ( 7g ) 1 + 175g + 360g 3 45y t β g3 = 7g 3 3 + 1 1 ( (4π) 10 g3 3 11g ) 1 + 45g 60g 3 0y t β gx = 43 6 g3 X 1 (4π) 59 6 g5 X β yt = y t ( 9 y t 17 ( β Yσ = 4 Y σ Tr Y σy σ β λh = 6y 4 t + 4λ h + λ h 0 g 1 9 4 g 8g 3 ) + 1 Y σy σ Yσ ) ( 1y t 9 ) 5 g 1 9g + 7 00 g4 1 + 9 0 g 1 g + 9 8 g4 + λ hφ + 1 λ hσ β λφ = 9/8 g 4 X 9g X λ φ + 4λ φ + λ hφ + 1/ λ φσ ( β λσ = 64Tr Y σy σ YσY ) ( σ + 16λ σ Tr Y σy ) σ + 18λ σ + λ hσ + λ φσ β λhφ = λ hφ (6y t + 1λ h + 1λ φ 4λ hφ 9/10 g 1 9/ g 9/ g X ( β λφσ = λ φσ (8Tr Y σy ) σ + 1λ φ + 6λ σ 4λ φσ 9/ g ) X + 4λ hσ λ hφ ) + λ hσ λ φσ ( β λhσ = λ hσ 6y t (Y + 8Tr σy ) σ + 1λ h + 6λ σ + 4λ hσ 9/10 g ) 1 9/ g + 4λ hφ λ φσ Alexandros Karam (Univeristy of Ioannina) 1508.03031 September 7, 015 1 / 8
Parameter Space Scan v h v φ v σ λ h (Λ) λ φ (Λ) λ σ(λ) λ hφ (Λ) λ φσ (Λ) λ hσ (Λ) M h3 46 11 770 0.176 0.004 0.57 0.0036 0.06 0.001 550.6 50 M N 40 GeV, M X 75 GeV 00 900 10 1 Λ Σ MN [GeV] 150 100 700 500 300 M h Mass [GeV] scalar couplings 10 Λ hσ Λ h Λ hφ Λ φσ 50 100 Λ φ 500 1000 1500 000 500 MX [GeV] 10 4 10 6 10 8 10 10 10 1 10 14 10 16 10 18 Μ GeV Y σ (M N ) = M N /v σ 0.31, g X (M X ) = M X /v φ.51 Alexandros Karam (Univeristy of Ioannina) 1508.03031 September 7, 015 / 8
DM Annihilations and Semi-annihilations Alexandros Karam (Univeristy of Ioannina) 1508.03031 September 7, 015 3 / 8
Boltzmann Equation The dark gauge bosons X a are stable due to a remnant SO(3) symmetry. The Boltzmann equation has the form dn dt + 3 H n = σv ( a n n ) σv eq s n (n n eq ) 3 3 M N = 40 GeV 10-4 10-5 10-6 10-7 (cm 3 / s) 10-8 10-9 10-30 10-31 10-3 10-33 10-34 <συ> a <συ> s 0 100 00 300 400 500 600 700 800 900 1000 M X [GeV] Alexandros Karam (Univeristy of Ioannina) 1508.03031 September 7, 015 4 / 8
Relic Density Ω X h = 3 1.07 109 GeV 1 g M P J(x f ), J(x f ) = x f dx σv a + σv s x Ω DM h = 0.1187 ± 0.0017 M X 700 750 GeV 50 00 900 MN [GeV] 150 100 700 500 300 M h Mass [GeV] 50 100 500 1000 1500 000 500 MX [GeV] Alexandros Karam (Univeristy of Ioannina) 1508.03031 September 7, 015 5 / 8
Direct Detection σsi X Ri R1i µred = fn MX mn πvh vφ Mhi i 10-43 σsi[cm ] 10-44 10-45 10-46 LUX (013) XENON 1T 10-47 0 00 400 600 800 1000 100 1400 1600 1800 000 MX [GeV] Alexandros Karam (Univeristy of Ioannina) 1508.03031 September 7, 015 6 / 8
Conclusions Classically scale invariant models do not suffer from the hierarchy problem and are very minimal BSM extensions Dark matter and neutrino mass scales are dynamically generated The model predicts three scalar bosons (one identified with 15 GeV Higgs) Stable vacuum Contains weak scale RH Majorana neutrinos TeV scale DM in agreement with direct detection limits Alexandros Karam (Univeristy of Ioannina) 1508.03031 September 7, 015 7 / 8
Thank you! Alexandros Karam (Univeristy of Ioannina) 1508.03031 September 7, 015 8 / 8
Effective Lagrangian Effective Lagrangian that contains all the interactions between the scalars and the rest of the fields with L h i eff = R i1h i ( M W v h W + µ W µ + M Z v h Z µz µ Mt v h tt M b v h bb Mc v h cc Mτ v h ττ + αs 1πv h G a µνg aµν + α πv h A µνa µν 3M X + R ih i XµX a aµ M N R i3h i N N + V ijk h ih jh k, v φ v σ V h ijk = R i1 [λ hφ R j (v h R k + v φ R k1 ) λ hσ R j3 (v h R k3 + v σr k1 ) + R j1 ( 6λ h v h R k1 + λ hφ v φ R k λ hσ v σr k3 )] + R i [λ hφ R j1 (v h R k + v φ R k1 ) λ φσ R j3 (v φ R k3 + v σr k ) + R j ( 6λ φ v φ R k + λ hφ v h R k1 λ φσ v σr k3 )] + R i3 [ λ hσ R j1 (v h R k3 + v σr k1 ) + λ φσ R j (v φ R k3 + v σr k ) + R j3 ( 6λ σv σr k3 λ hσ v h R k1 + λ φσ v φ R k )] ) Alexandros Karam (Univeristy of Ioannina) 1508.03031 September 7, 015 8 / 8
Higgs Total Decay Width and Signal Strength Parameter Total decay width Γ tot h 1 of a SM Higgs-like scalar h 1 with M h1 = 15.09 GeV: Γ tot h 1 = cos α cos β Γ SM h 1 + Γ inv h 1, Γ inv h 1 = Γ (h 1 XX) + Γ ( h 1 NN ), Γ (h 1 h h ) = Γ (h 1 h 3 h 3 ) = 0 Construct the signal strength parameter µ h1 : It simplifies to: which translates to µ h1 = σ (pp h 1) BR (h 1 χχ) σ SM (pp h) BR SM (h χχ) = cos4 α cos 4 β ΓSM h1 Γ tot Using the first benchmark set of values: µ h1 cos α cos β > 0.81, @ 95% CL R 11 = cos α cos β > 0.9. R 11 = 0.994, which lies comfortably within the allowed range. Alexandros Karam (Univeristy of Ioannina) 1508.03031 September 7, 015 8 / 8 h 1