Constraints from the renormalisation of the minimal dark matter model

Transcription

1 Constraints from the renormalisation of the minimal dark matter model James McKay Imperial College London in collaboration with Peter Athron and Pat Scott July 20, 2016

2 Outline 1. Minimal dark matter 2. RGE running 3. Numerical calculation of electroweak mass splitting phenomenological impact how to calculate a mass understanding the origin of the problem

3 Minimal dark matter L = L SM χ (i /D M) χ where /D is the SU(2) L covariant derivative. Proposed by Cirelli, Fornengo and Strumia [Cirelli et al., 2006, Cirelli and Strumia, 2009]. Fermionic SU(2) L quintuplet, χ, with hypercharge Y = 0 with interactions: ( L int =g χ + γ µ χ + + 2χ ++ γ µ χ ++) (s w A µ + c w Z µ ) + g ( 3χ + γ µ χ 0 + 2χ ++ γ µ χ +) W + µ + h.c. One new parameter, the MDM tree-level degenerate mass, M

4 Minimal dark matter phenomenology Relic density: M 9.6 ± 0.2 TeV by default has a non-zero mass splitting M = Mχ M χ 0 between charged and neutral component

5 Minimal dark matter phenomenology Relic density: M 9.6 ± 0.2 TeV by default has a non-zero mass splitting M = Mχ M χ 0 between charged and neutral component Not yet ruled out by direct detection Collider searches for charged particles also depends on mass splitting

6 Minimal dark matter phenomenology Relic density: M 9.6 ± 0.2 TeV by default has a non-zero mass splitting M = Mχ M χ 0 between charged and neutral component Not yet ruled out by direct detection Collider searches for charged particles also depends on mass splitting Vacuum stability Perturbitivity

7 Electroweak vacuum stability The standard model Higgs potential is not absolutely stable V We are here Is nice to make it absolutely stable and make sure is not absolutely unstable We must renormalise, V = V (µ), to quantify this instability Could we end up here? µ V = mh 2 (µ) H 2 + λ(µ) H 4 for µ M Z we use µ O(h) V λ(h)h 4

8 Electroweak vacuum stability The standard model Higgs potential is not absolutely stable V We are here Is nice to make it absolutely stable and make sure is not absolutely unstable We must renormalise, V = V (µ), to quantify this instability Could we end up here? V = m 2 H (µ) H 2 + λ(µ) H 4 for µ M Z we use µ O(h) V λ(h)h 4 Problem if λ < 0 as h µ

9 3 types of stability Stable one global minimum Scalar dark matter, Minimal dark matter... Meta-stable but expected time before bubble nucleation age of universe the standard model Unstable time before bubble nucleation 0 hopefully not your new model!

10 Running of λ(µ) in MDM model 0.10 M t = ±0.3 GeV (- - 2σ) α S = ± (.. 3σ) O(TC) = 1.0 O(β) = 1.0 M H = ±0.3 GeV (-. 3σ) λ(µ) M χ0 = 9.8 TeV Metastable Full 2-loop RGEs solved using FlexibleSUSY with χ integrated in at M Z 0.05 Unstable µ (GeV) 0.10 M t = ±0.3 GeV (- - 2σ) O(TC) = 1.0 α S = ± (.. 3σ) O(β) = 2.0 M H = ±0.3 GeV (-. 3σ) M χ0 = 9.8 TeV M t = ±0.3 GeV (- - 2σ) O(TC) = 0.0 λ(µ) α S = ± (.. 3σ) M H = ±0.3 GeV (-. 3σ) O(β) = M χ0 = 9.8 TeV Metastable 0.05 λ(µ) 0.05 Unstable µ (GeV) Metastable 0.05 Unstable µ (GeV)

11 Perturbitivity Landau poles [Cai et al., 2015] showed MDM septuplet has Landau pole at 10 8 GeV

12 Perturbitivity Landau poles [Cai et al., 2015] showed MDM septuplet has Landau pole at 10 8 GeV Most recent analysis [Di Luzio et al., 2015] uses partial 2-loop RGEs MDM Landau pole at GeV. Using full 2-loop RGEs we find: SM GeV MDM GeV λ λ 1.6 g g g g λ(µ) λ(µ) µ (GeV) µ (GeV)

13 Spectrum generators Spectrum generator tool which calculates particle masses and couplings given known parameters Can solve renormalisation group equations (RGEs)

14 Spectrum generators Spectrum generator tool which calculates particle masses and couplings given known parameters Can solve renormalisation group equations (RGEs) e.g. FlexibleSUSY, SPheno, SOFTSUSY not only for supersymmetric models BUT while these are amazing tools we can t always use them blindly the following part of this talk will demonstrate an example of why

15 Electroweak multiplet mass splitting Radiatively induced mass splittings essential for MDM and wino-like dark matter in SUSY M = Mχ M χ MeV for M M Z [Cirelli et al., 2006, Ibe et al., 2013, Yamada, 2010, Del Nobile et al., 2010, Ostdiek, 2015] So any spectrum generator should reproduce this result.

16 Spectrum generator results 200 SPheno FlexibleSUSY M (Mev) 100 m (Gev) Degenerate mass M (GeV) Degenerate mass (GeV)

17 Electroweak multiplet mass splitting The decoupling between states vanishes for large M depending on how the radiative corrections are calculated Explicit pole mass Implicit pole mass M χ + + M χ M (Mev) M χ + M χ Degenerate mass M (GeV)

18 How important is it? Lifetime of χ The decay width for the electron channel is and Γ µ = 0.12Γ e Γ e = (n 2 1) G 2 F M5 60π Explicit pole mass Implicit pole mass 10 8 τ (mm/c) Degenerate mass M (GeV)

19 So... how do we calculate a pole mass? Find the pole of the propagator: where Σ X = Σ (1) X /p M 0 + Σ K (p 2 )/p + Σ M (p 2 ) = 0 + Σ(2) X +...

20 So... how do we calculate a pole mass? setting p = M pole to be the pole mass we have [ M0 Σ M (Mpole 2 ) ] now just iterate, easy! M pole = Re 1 + Σ K (M 2 pole ). [ ] M pole Re M 0 Σ (1) M (M2 pole ) M 0 Σ (1) K (M2 pole )

21 So... how do we calculate a pole mass? setting p = M pole to be the pole mass we have [ M0 Σ M (Mpole 2 ) ] now just iterate, easy! M pole = Re 1 + Σ K (M 2 pole ). [ ] M pole Re M 0 Σ (1) M (M2 pole ) M 0 Σ (1) K (M2 pole ) lim M M 0 MeV

22 Explicit pole mass Approximate! Σ (1) M (M2 pole ) Σ(1) M 2M 0 (M 0 Σ (1) K + Σ(1) M ) Σ (1) M [ ] M pole Re M 0 Σ (1) M (M2 0 ) M 0 Σ (1) K (M2 0 ) p 2 =M 2 0

23 Explicit pole mass Approximate! Σ (1) M (M2 pole ) Σ(1) M 2M 0 (M 0 Σ (1) K + Σ(1) M ) Σ (1) M [ ] M pole Re M 0 Σ (1) M (M2 0 ) M 0 Σ (1) K (M2 0 ) p 2 =M 2 0 lim M M 166 MeV

24 How to calculate a pole mass? So what? They are equivalent expressions to the same order in the gauge coupling. But remember, O(M pole ) = 1 TeV, errors are large, but O( M) = 100 MeV. Have to look more closely at the self energy functions.

25 The electroweak multiplet self energy W Z/A χ 0 χ 0 g g χ ± χ ± g g χ ± χ ±

26 The electroweak multiplet self energy Neutral component, W boson only Σ χ0 M (p2 ) = 3g 2 2π 2 ( B0 (p 2, M 2, m 2 W ) 1) Σ χ0 K (p2 ) = 3g 2 4π 2 ( B1 (p 2, M 2, m 2 W )) Charged component, W, A and Z Σ χ M (p2 ) = g 2 M ( ) s 2 4π 2 W B 0(p 2, M 2, 0) + cw 2 B 0(p 2, M 2, mz 2 ) + 5B 0(p 2, M 2, mw 2 ) 3 Σ χ K (p2 ) = g 2 M ( ) s 2 8π 2 W B 1(p 2, M 2, 0) + cw 2 B 1(p 2, M 2, mz 2 ) + 5B 1(p 2, M 2, mw 2 )

27 Explicit pole mass result M = Mχ M χ 0 [ ] = M Σ χ (M, M) = Σ χ0 (M, M) Σ χ (M, M) [ ] M Σ χ0 (M, M) M (B 0 (M, M, m z ) B 0 (M, M, m w )) +...

28 Explicit pole mass result But for M m Z M = Mχ M χ 0 [ ] = M Σ χ (M, M) = Σ χ0 (M, M) Σ χ (M, M) [ ] M Σ χ0 (M, M) M (B 0 (M, M, m z ) B 0 (M, M, m w )) +... B 0 (M, M, m) πm/m + terms which will cancel

29 Explicit pole mass result But for M m Z M = Mχ M χ 0 [ ] = M Σ χ (M, M) = Σ χ0 (M, M) Σ χ (M, M) [ ] M Σ χ0 (M, M) M (B 0 (M, M, m z ) B 0 (M, M, m w )) +... B 0 (M, M, m) πm/m + terms which will cancel So we end up with M α 2 2 ( mw c 2 W m Z )

30 What is causing this difference? Σ(M 2 pole ) = Σ(M2 pole, M2, m 2 W ) Σ(M2, M 2, m 2 W ) = Σ(M2 ) B n (M 2 pole, M2, m 2 W ) B n(m 2, M 2, m 2 W ) Some kind of threshold problem in the Passarino-Veltman functions

31 Difference of Passarino-Veltmann functions M π (B n(rm, M, m 1 ) B n (rm, M, m 2 )) M π (B 0(rM, M, m 1 ) B 0 (rm, M, m 2 )) r = 1 r = r = r = 0. 9 r = M

32 So which method is right? At 1-loop both are finite and equivalent to the same loop order Need to go to 2-loop order At 2-loop iterative method is IR divergent due to photons yet explicit method has cancellation of IR divergences from derivates of 1-loop functions [Ibe et al., 2013]

33 So which method is right? At 1-loop both are finite and equivalent to the same loop order Need to go to 2-loop order At 2-loop iterative method is IR divergent due to photons yet explicit method has cancellation of IR divergences from derivates of 1-loop functions [Ibe et al., 2013] Σ (1) (Mpole 2 ) Σ(1) + C 1 Σ (1) + C 1 Σ (1) + O(g 6 ) Σ (2) (Mpole 2 ) Σ(2) + Σ (2) + O(g 6 ) p 2 =M 2 0 p 2 =M 2 0

34 Summary Provided full two-loop results for vacuum stability and perturbativity Important problem in the numerical calculation of pole masses with implications for MDM and Wino dark matter Publication in preparation including the results presented here and two-loop self energies

35 References Athron, P., Park, J.-h., Stöckinger, D., and Voigt, A. (2014). FlexibleSUSY A spectrum generator generator for supersymmetric models. Computer Physics Communications, 190:45. Cai, C., Huang, Z. M., Kang, Z., Yu, Z. H., and Zhang, H. H. (2015). Perturbativity limits for scalar minimal dark matter with Yukawa interactions: Septuplet. Physical Review D - Particles, Fields, Gravitation and Cosmology, 92(11):1 15. Cirelli, M., Fornengo, N., and Strumia, A. (2006). Minimal dark matter. Nuclear Physics B, 753(1-2): Cirelli, M. and Strumia, A. (2009). Minimal dark matter: Model and results. New Journal of Physics, 11: Del Nobile, E., Franceschini, R., Pappadopulo, D., and Strumia, A. (2010). Minimal matter at the large hadron collider. Nuclear Physics B, 826(1): Di Luzio, L., Grober, R., Kamenik, J. F., and Nardecchia, M. (2015). Accidental matter at the LHC. page 52. Ibe, M., Matsumoto, S., and Sato, R. (2013). Mass splitting between charged and neutral winos at two-loop level. Physics Letters, Section B: Nuclear, Elementary Particle and High-Energy Physics, 721(4-5): Ostdiek, B. (2015). Constraining the minimal dark matter fiveplet with LHC searches. (2):8.

36 Additional slides

37 The probability of quantum tunnelling probability of decay A e B/ S E 2π 2 ρ 3 dρ 0 (T uλ B ) 4 e S E. [ 1 2 ( ) ] 2 dφ + V dρ where ρ = ( τ 2 + x 2) 1/2 and φ satisfies the Euclidean equation of motion d 2 φ dρ + 3 dφ 2 ρ dρ = dv dφ lim ρ φ(ρ) = q 0, t, the global vacuum at t = 0 and back to start at t. ( p V(q) 0 e 140 Λ B M Planck a q S E = 8π2 3 λ b ) 4 ( exp 26 ) λ min The probability that <number of events> = 0 is L = exp [ p]

38 RGE running in MDM model At the one-loop level and at the two-loop level β (1) g 2,SM = 19 6 g 3 2 β (1) g 2,MDM = 7 2 g 3 2 (1) β (2) g 2,SM = 1 30 g 3 2 (360g Tr(Y d Y d ) 45Tr(Y uy u ) 15Tr(Y e Y e ) + 27g g 5 2 ) (2) is modified as (using SARAH to generate RGEs) β (2) g 2,MDM = β(2) g 2,SM g 5 2. (3)

39 Consider a scalar field φ, for bubble nucleation in flat 4-dimensional spacetime we can take φ to be a function of ρ = ( τ 2 + x 2) 1/2 only, as there exists an O(4) symmetry. B is given by the action B = S E 2π 2 ρ 3 dρ 0 [ 1 2 ( ) ] dφ 2 + V dρ for φ satisfying the Euclidean equation of motion (4) d 2 φ dρ dφ ρ dρ = dv dφ (5) for which the boundary conditions can be combined into the single requirement that lim ρ φ(ρ) = q 0, the solution to (??) is known as the bounce solution, as it describes, in Euclidean time, a solution which is in the unstable vacuum at t, the global vacuum at t = 0 and bounces back to the starting point at t.

40 h(ρ) = 2 2R λ ρ 2 + R 2 (6) and from we obtain the action S E = 8π2 3 λ (7) Γ (T u Λ B ) 4 e S E. (8)

41 The probability of quantum tunnelling probability of decay A e B/ B relatively easy to calculate from potential A much more difficult but can make good approximations on dimensional grounds

42 The likelihood function Typically called the probability, which is interpreted as the expected number of decay events ( p e 140 Λ B M Planck ) 4 ( exp 26 ) λ min The probability that <number of events> = 0 is L = exp [ p]

43 The number of decay events/ log of the likelihood M H = ±0.3 (1σ band -- ) α S = L 0 log 10 ( log(l)) L Top pole mass (Gev)

44 Stability regions in the SM Right: Figure from Degrassi, G., Di Vita, S., Elias-Miro et al M t (Gev) 180 Instability Metastability Stability M H (Gev) Pole top mass Mt in GeV Instability ,2,3 Σ Meta stability Stability h Bound for absolute vacuum stability ( Mt M H > ) 0.6 ( ) αs