MIAMI 2012 December 13-20, 2012, Fort Lauderdale, Florida, USA

Transcription

1 NEWS FROM LHC: ARE WE LIVING IN A METASTABLE VACUUM? Vincenzo Branchina and Emanuele Messina Department of Physics, University of Catania, Italy and INFN, Sezione di Catania, Italy MIAMI 01 December 13-0, 01, Fort Lauderdale, Florida, USA 1

2 It is widely accepted that the top loop-corrections to the Higgs effective potential destabilize the electroweak vacuum... NOT IN SCALE Instability E W

3 One-Loop Effective Potential V 1l (φ) NOT IN SCALE Instability E W [ ( V 1l (φ) = 1 m φ + λ 4 φ π m + λ ) ( ( ) ) m + λ φ ln φ µ 3 +3 (m + λ6 ) ( ( ) ) m + λ φ 6 ln φ µ g 1 4 ( ( 1 16 φ4 4 ln g 1 φ ) µ 5 ) 6 ( g1 + g ) ( 1 ( +3 φ (ln 4 4 g1 + g ) ) ) φ 16 µ 5 ( 1 h 4t φ 4 ln g φ 6 µ 3 ) ] 3

4 RG Improved Effective Potential V RGI (φ) NOT IN SCALE Instability E W V RGI (φ) = m (t) φ (t) + λ eff (t) φ4 (t) 4 ( t 1 ln φ µ ) [ λ eff (t) = λ g 1 8π 8 ( ln ( g1 with λ = λ(t), g = g(t), g 1 = g 1 (t), g = g (t). 4 ) 5 ) ( 1 h 4 t ln g 3 )+ 3 ( g 1 + g ) 6 16 ( ln ( ) g1 + g 5 ) ] 4 6 4

5 This instability occurs for large values of the field V RGI (φ) well approximated by keeping only the quartic term : V RGI (φ) λ eff(φ) φ 4 4 and λ eff (φ) depends on φ essentially as λ(µ) depends on µ Read the Effective Potential from the λ(µ) flow... and explore the possibility that SM valid up to very high scales... Planck scale???... But ignore the Naturalness Problem!!!... 5

6 Running of λ(µ) in the SM From: Degrassi, Di Vita, Elias-Miro, Espinosa, Giudice, Isidori, Strumia, JHEP 108 (01) Higgs quartic coupling Λ Μ M h 15 GeV 3Σ bands in M t ± 0.7 GeV Α s M Z ± M t GeV Α s M Z Α s M Z M t GeV Μ Blue thick line : M H = 15 GeV, m t = GeV ; λ(µ) = 0 for µ GeV... Going to read the Effective Potential from the λ(µ) flow... 6

7 NOT IN SCALE Instability E W 7

8 Stability Criterium I Highest possible value Λ max for the physical cut-off Λ (= highest possible value for the scale of new physics) of the Standard Model... 1) EW minimum v : V Higgs RGI (v) = 0 ) For large values of φ : V Higgs RGI (φ) λ eff (φ) 4 φ 4 3) λ eff (φ) behaves with φ as λ(µ) with µ. 4) V Higgs RGI (φ max ) = 0 : Λ max = φ max i.e.: Λ max from λ(µ) through λ(λ max ) = 0 Λ max as the highest possible (limiting) value of Λ In order for the theory to be stable : λ(λ) > 0 This criterium can be somehow softened... pushing Λ max to higher values... (again not paying attention to Naturalness) 8

9 Stability Criterium II : Meta-stability Λ Μ log 10 Μ GeV Running of λ(µ) It should be possible to accept negative values of λ(µ) beyond λ(µ 0 ) = 0 if later on (for µ > µ 0 ) the potential is stabilized through one of the following mechanisms. 1) λ(µ) reaches again positive values ; ) the potential is stabilized by the presence of higher order terms (new physics at the Planck scale) This leads some people to propose the Metastability Scenario 9

10 Meta-stability scenario NOT IN SCALE Instability E W Vacuum Decay 10

11 Some References on Instability N. Cabibbo, L. Maiani, G. Parisi, R. Petronzio, Nucl.Phys. B158 (1979) 95. R.A. Flores, M. Sher, Phys. Rev. D7 (1983) M. Lindner, Z. Phys. 31 (1986) 95. M. Sher, Phys. Rep. 179 (1989) 73. M. Lindner, M. Sher, H. W. Zaglauer, Phys. Lett. B8 (1989) 139. C. Ford, D.R.T. Jones, P.W. Stephenson, M.B. Einhorn, Nucl.Phys. B395 (1993) 17. M. Sher, Phys. Lett. B317 (1993) 159. G. Altarelli, G. Isidori, Phys. Lett. B337 (1994) 141. J.A. Casas, J.R. Espinosa, M. Quirós, Phys. Lett. B34 (1995) 171. J.A. Casas, J.R. Espinosa, M. Quirós, Phys. Lett. B38 (1996)

12 Some References on Meta-stability D.L. Bennett, H.B. Nielsen and I. Picek, Phys. Lett. B 08 (1988) 75. G. Anderson, Phys. Lett. B43 (1990) 65 P. Arnold and S. Vokos, Phys. Rev. D44 (1991) 360 J.R. Espinosa, M. Quiros, Phys.Lett. B353 (1995) C. D. Froggatt and H. B. Nielsen, Phys. Lett. B 368 (1996) 96. C.D. Froggatt, H. B. Nielsen, Y. Takanishi (Bohr Inst.), Phys.Rev. D64 (001) G. Isidori, G. Ridolfi, A. Strumia, Nucl. Phys. B609 (001) 387. J. Elias-Miro, J. R. Espinosa, G. F. Giudice, G. Isidori, A. Riotto, A. Strumia, Phys.Lett. B709 (01) -8 G. Degrassi, S. Di Vita, J. Elias-Miro, J. R. Espinosa, G. F. Giudice, G.Isidori, A. Strumia, JHEP 108 (01)

13 Before we begin studying this instability issue we might learn something by considering the simpler Higgs-Yukawa model 13

14 Higgs-Yukawa model Euclidean model in d = 4 dimensions with N fermion flavors L = N i=1 ψ i ( / + gφ ) ψ i + 1 ( µφ) + m φ + λ 4! φ4 14

15 Leading order in the 1/N-expansion (Λ = momentum cut-off ; rescaling ϕ = g φ ) V (ϕ) = m g ϕ + λ 4!g 4 ϕ4 N 16π ( ) Λ 4 ln (1 + ϕ Λ + Λ ϕ ϕ 4 ln )) (1 + Λ ϕ ϕ Λ << 1 V (ϕ) = m g ϕ + τ = m g V (ϕ) = τ ϕ + 1 4! λ 4!g 4 ϕ4 N 16π N 4π Λ ( λ g 4 3N π ( ( ) ϕ Λ ϕ + ϕ 4 ln Λ 1 ) ϕ4 ( ( ) ϕ ln Λ 1 )) ϕ 4 Let us look for minima (and maxima) of V (ϕ) and investigate the stability issue... ( dv dϕ = ϕ τ + 1 λ 6 g 4 ϕ N ) 4π ϕ ln ϕ Λ = 0 Intrestingly we find analytical solutions... 15

16 Max.: ϕ = 0 ; Min.: ϕ min Λ = W 1 g τ/m c g τ m e π c 3N λ g 4! ; Max.: ϕ max Λ = W 0 g τ m c g τ/m c exp π 3N «λ g V eff φ τ in the interval : g e π 3N m c For τ 0, W 0 (x) can be expanded : ϕ ( max π Λ = exp 3N λ g 4 1 < τ < 0 ; W 0 (x) and W 1 (x) : Lambert functions. ) λ g 4 + g m c τ + O ( τ ) 16

17 ϕ max Λ ( π = exp 3N ) λ + g τ g 4 m c + O ( τ ) Bare coupling λ = λ(λ) : defines the model at the UV scale Λ, the physical cut-off. It is the running coupling at this scale. Beyond Λ, UV completion needed. From the physical requirement of stability : λ = λ(λ) 0. Maxima at values of ϕ/λ which do not respect the condition ϕ/λ << 1. Expansion considered for V (ϕ) no longer valid. When ϕ/λ << 1 Scaling Region Continuum Limit, Renormalized Theory. Maxima beyond scaling region! Around and beyond maxima : Expansion for V (ϕ) no trustable. Moreover : In the scaling region V (ϕ) = τ ϕ + 1 4! ( λ g 4 3N π ( ( ) ϕ ln Λ 1 )) ϕ 4 is the Renormalized Effective Potential 17

18 Renormalized Effective Potential Defining the renormalized quantities τ R, λ R, g R, ϕ R, ψ R from : Z ϕ τ = τ R + δτ ; Z φ λ g 4 = λ R g 4 R + δ λ g 4 ; 1 g Z ϕ = 1 g R where Z ϕ = 1 + δz ϕ and Z ψ = 1 + δz ψ + δ 1 g ; ϕ = Z 1/ ϕ ϕ R ; ψ = Z 1/ ψ and choosing the renormalization conditions with location of the minimum and second derivative of V (ϕ) as for the tree level : ψ R δτ = 0 ; δ 1 g = N 8π ln µ Λ ; δz ψ = 0 ; δz ϕ = 0 ; δ λ g 4 = 3N π ln µ Λ We have : τ R = τ ; ϕ R = ϕ ; λ R g 4 R = λ g 4 3N π ln µ Λ Therefore : V (ϕ) = τ ϕ + 1 ( λ 4! g 4 3N π = τ R ϕ R + 1 ( λr 4! gr 4 3N π In other words... ( ( ϕ ln ( ln Λ ) 1 ) 1 ( ϕ R µ )) ϕ 4 )) ϕ 4 R. 18

19 Renormalized Effective Potential Renormalized Effective Potential in terms of the bare parameters V (ϕ) = τ ϕ + 1 ( λ 4! g 4 3N ( ( ) ϕ π ln Λ 1 )) ϕ 4 The same Renormalized Effective Potential in terms of the renormalized parameters V (ϕ) = V (ϕ R ) = τ R ϕ R + 1 ( λr 3N ( ( ) ϕ 4! π ln R µ 1 )) ϕ 4 R. V eff g 4 R φ 19

20 Renormalized Effective Potential V eff Bare param.: V (ϕ) = τ ϕ + 1 4! φ ( λ g 4 ( ) 3N π (ln ϕ Λ 1 Reno. param.: V (ϕ) = V (ϕ R ) = τ R ϕ R + 1 4! ( λr g 4 R Bare Potential before expansion in ϕ/λ : )) ϕ 4 3N π (ln ( ) )) ϕ R µ 1 ϕ 4 R (dashed line) (dashed line) (continuum line) Lesson : If we only know the Effective Potential V (ϕ R ) written in terms of the renormalized parameters, we cannot see that for values of ϕ R beyond a certain value this expression is no longer valid! V (ϕ R ) ceases to be the renormalized potential because we are beyond the scaling region! In Dim. Reg., we only have access to the effective potential written as in the second line. Information contained in the comparison between first and third line : lost! 0

21 Renormalization Group Improved Bare Potential Callan-Symanzick equation for the bare potential : [ Λ Λ + β 1/g 1/g + β λ/g 4 (λ/g 4 ) γ ϕ τ τ γ ϕϕ ϕ Functions τ(t), λ g 4 (t) and ϕ(t) solutions of the RG equations d dt λ dτ g = β 4 λ/g 4 ; dt = γ ϕ τ ; dϕ dt = γ λ ϕϕ with : g (t = 0) = λ 4 g 4 ] V (ϕ, τ, 1g, λg ) 4, Λ = 0 ( )) t = 1 (ln ϕ Λ 1 ( V RGI τ(t), λ ) g 4 (t), ϕ(t) = 1 τ(t)ϕ(t) + 1 λ 4! g 4 (t)ϕ4 (t) : ; τ(t = 0) = τ ; ϕ(t = 0) = ϕ RG functions, leading order in 1/N : β λ/g 4 = 3N π ; β 1/g = N 4π ; γ ϕ = 0 ; γ ϕ = 0 λ g 4 (t) = λ g 4 3N π t ; τ(t) = const. = τ ; ϕ(t) = const. = ϕ V RGI V (ϕ) = τ ϕ + 1 4! ( λ g 4 3N π ( ( ) ϕ ln Λ 1 )) ϕ 4 There is no RG improvement since in the leading order of the 1/N expansion the theory is exactly solvable. 1

22 Renormalization Group Improved Renormalized Potential Callan-Symanzick equation for the renormalized potential: [ µ µ + β 1/gR (1/gR ) + β λ R /gr 4 (λ R /gr 4 ) γ ϕ τ R γ ϕr ϕ R R τ R ϕ R with : t = 1 Finally : V RGI (τ R, ( ) ln ϕ R µ 1 1 g R, λ ) R gr 4, ϕ R, µ = 1 τ R(t)ϕ R (t) + 1 4! V RGI V (ϕ) = V (ϕ R ) = τ R ϕ R + 1 ( λr 4! gr 4 3N π ( ln ] V λ R gr 4 (t)ϕ R (t) 4 ( ϕ R, τ R, ( ) ϕ R µ 1 )) ϕ 4 R 1 g R, λ ) R gr 4, µ = 0

23 Back to the Standard Model 3

24 Higgs Effective potential Large values of φ : V Higgs RGI (φ) λ eff (φ) 4 φ 4 and λ eff (φ) λ(µ) We are interested in the running of λ(µ), more generally in the running of the SM couplings (coupled RG equations) 4

25 RG equations for the SM couplings µ d dµ λ(µ) = β λ (λ, h t, {g i }) µ d dµ h t(µ) = β ht (λ, h t, {g i }) µ d dµ g i(µ) = β gi (λ, h t, {g i }) with i = 1,, 3 and g i = {g, g, g s } 5

26 Running of the SM couplings SM couplings log 10 Μ GeV M t = GeV, M H = 15 GeV, M Z = GeV, λ(m t ) = 4.53, h t (M t ) = 0.936, g (M Z ) = 0.65, 5/3g(M Z ) = 0.46, g s (M Z ) = 1.. λ(µ) (black line), h t (µ) (red, solid line), g (µ) (blue line), g(µ) (greenline) g s (µ) (red, dashed line). Let us focus on λ(µ) 6

27 Running of λ(µ) Λ Μ Λ Μ log 10 Μ GeV log 10 Μ GeV 7

28 ...Higgs Effective Potential NOT IN SCALE Instability E W New Minimum Can we trust the formation of the non-trivial Maximum? Can we trust the Instability Region? Can we trust the formation of the New Minimum? Possibly Yes Possibly Yes Not so obvious... This is the form of the effective potential that leads some people to propose the Meta-Stability scenario... 8

29 Meta-stability Scenario NOT IN SCALE Instability E W Vacuum Decay The idea is to consider the tunneling between the EW vacuum and the true vacuum. As long as the tunneling time is larger than the age of the Universe... we can live in a Meta-Stable (EW) Vacuum... How do we compute the tunneling probability? 9

30 Tunneling and bounces Bounce : solution to Euclidean equations of motion (radial coordinate) µ µ φ + d V (φ) d φ = φ d dr 3 dφ r dr + d V (φ) d φ = 0, Boundary conditions : φ (0) = 0, φ( ) = v 0. The authors choose : V (φ) = λ 4 φ4 with constant and negative λ : λ < 0!!! Bounce solution : φ 0 (r) = R, S[φ λ r + R 0 ] = 8π 3 λ, R is the size of the bounce (degeneracy, zero mode) Fluctuation determinant : Zero modes related to the location of the bounce (Volume factor), to the size of the bounce (integration in R)... Jacobian factor related to the zero modes (R 4 factor) 30

31 Probability for the decay into the new vacuum : ( µ 1 R ) p = max R ] V U [ R exp 8π 4 3 λ(µ) S tunnelling rate /R in GeV Picture : G. Isidori, G. Ridolfi, A. Strumia, Nucl.Phys.B 609 (001)

32 Tunneling From the decay probability the EW vacuum lifetime τ is computed. If τ T U (T U age of the universe), an EW metastable vacuum is acceptable We might live in a metastable vacuum... But : τ is larger, equal or smaller than T U depending on the values of the SM parameters The following picture in terms of M H and m t is then obtained 3

33 A broader picture : Stability - Meta-stability - Instability Pictures : Degrassi, Di Vita, Elias-Miro, Espinosa, Giudice, Isidori, Strumia, JHEP 108 (01) Instability Top mass Mt in GeV Meta stability Stability Non perturbativity Pole top mass Mt in GeV Instability ,,3 Σ Meta stability Stability h h Central (preferred experimental) values : M H = 15 GeV and m t = GeV 33

34 Stability - Meta-stability - Instability NOT IN SCALE E W Vacuum Decay Instability Pole top mass Mt in GeV Instability ,,3 Σ Meta stability Stability h... However... 34

35 From the Running of λ(µ) with M H = 15, m t = NOT IN SCALE Instability = GeV M P E W = 46 GeV 10 9 GeV!!! New minimum at µ = 10 9 Gev!!!! Now in scale... 35

36 Effective Potential (M H = 15, m t = 173.1) Log-Log 100 sign V log 10 V GeV log 10 Φ GeV 36

37 We need higher order terms (new physics) to make the potential stable before (at least at) the Planck scale... NOT IN SCALE Instability = GeV M P E W = 46 GeV Higher Order Operators!! Is the presence of these new physics operators harmless? 37

38 New physics operators around the Planck scale Let us add λ 6 φ 6 : V (φ) = λ 4 φ ! One-loop β-function of λ 6 : λ 6 φ 6 MP β λ6 = λ π ( 360h 6 t 6480λ λ λ 6 ) + 6γφ λ 6 One-loop β-function of λ : β λ = 1 16π ( 4λ 6h 4 t ( g 4 + g g + 3g )) + 4γ φ λ + 1 λ 6 16π 6 with : γ φ = 1 16π ( 3h t 9 4 g 3 4 g) anomalous dimension of φ. V RGI is then : V RGI = 1 4 λ eff(φ)φ ! with: λ 6 (φ) λ 6 ( t = 1 ln φ µ ), ξ(φ) = e R 1 ln φ µ 0 dt γ φ λ 6 (φ) ξ(φ) 6 φ 6 MP... and here is the Log-Log plot 38

39 Higgs Effective Potential Log-Log (M H = 15, m t = 173.1) 150 sign V log 10 V GeV log 10 Φ GeV Blue line : Higgs Potential with no higher order term Red line : Higgs Potential with the higher order term λ 6 ϕ 6 MP (with λ 6 = 1 at µ = M P ). 39

40 Higgs Effective Potential (1-σ) Log-Log (M H = 15, m t = ) 150 sign V log 10 V GeV log 10 Φ GeV Blue line : Higgs Potential with no higher order term Red line : Higgs Potential with the higher order term λ 6 ϕ 6 MP (with λ 6 = 1 at µ = M P ). 40

41 Higgs Effective Potential (-σ) Log-Log (M H = 15, m t = ) 150 sign V log 10 V GeV log 10 Φ GeV Blue line : Higgs Potential with no higher order term Red line : Higgs Potential with the higher order term λ 6 ϕ 6 MP (with λ 6 = 1 at µ = M P ). 41

42 New physics operators and the bounce (tunneling) Equation for the bounce with no new physics operators : with has the solution: φ 0 (r) = Φ d φ dr 3 dφ r dr + V (φ) V (φ) = λ 4 φ4 λ R for λ < 0 r +R r Bounce with R = 10 and λ = 0.01 (Planck units) 4

43 Two new physics operators : V (φ) = λ 4 φ ! Φ r λ 6 M P φ ! Bounce without λ 6 and λ 8 (black). New bounce (red) In cut-off (Planck) units : λ 8 φ 8 MP 4 (λ = 0.01, λ 6 = 5, λ 8 = 0.5) size of the black bounce : R = 10 ; size of the red bounce : R = 1 Mini-bounces (size 1/Λ) saturate the path integral!!! 43

44 New physics operators and the bounce (tunneling) (M H = 15, m t = 173.1) And the Potential is safe at the Planck scale... 44

45 What about this picture??? Preferred experimental values : M H = 15 GeV and m t = GeV Pictures : Degrassi, Di Vita, Elias-Miro, Espinosa, Giudice, Isidori, Strumia, JHEP 108 (01) Instability Top mass Mt in GeV Meta stability Stability Non perturbativity Pole top mass Mt in GeV Instability ,,3 Σ Meta stability Stability h h... I think a bit in truble... a) λ negative...???... b) Strong impact of new physics operators... 45

46 A similar problem Higgs Inflation Scenario 46

47 ... Higgs Inflation Scenario... 3-σ from the central value m t = sign V V 1 4 M P Φ M P The potential without new physics operators for M H = 15 : m t = , , , , Enormous sensitivity to m t!!! 47

48 ... Higgs Inflation... 3-σ from the central value m t = Study the impact of the new physics operators sign V log 10 V GeV log 10 Φ GeV Potential for M H = 15 and m t = with no new physics operators (blue line) and with the inclusion of new physics operators (red line) (λ 6 = 1, , 0.5, 0.001, and λ 8 = 0, 1, 1, 1 from top to bottom) Enormous sensitivity to new physics operators too!!! What about Higgs Inflation???? 48

49 Conclusions We have to pay the due attention to irrelevant operators. Maybe they are not so irrelevant sometimes... They may cause the path integral to be saturated by mini-instantons or mini-bounces configurations, which change dramatically the tree-level picture. The conclusion that the experimental data today prefer a metastable EW vacuum does not seem on firm ground : it does not take into account the role of higher order (irrelevant) terms which are needed to make the potential stable at the Planck scale. The Higgs inflation scenario has similar problems. 49

50 Running of λ(µ) Λ Μ log 10 Μ GeV 50

51 Λ Μ log 10 Μ GeV 51

52 An instructive little excursion in d = 3 dimensions The bending down of the potential is due to the presence of the ϕ 4 term. In d = 4 dimensions, ϕ 4 is a marginal operator, while in d < 4 it is irrelevant. If we only keep relevant operators (scaling region), in d < 4 dimensions it should not be included. Let us consider now the d = 3 dimension case V = 1 m g ϕ + 1 λ 4! g 4 ϕ4 N 8π and keep anyhow the operator ϕ 4 : ( m V (ϕ) = g N ) Λ ϕ π + N 1π [ 4 9 Λ Λϕ 4 ( ) Λ 3 ϕ 3 ArcTan + ( ) ln (1 + ϕ ϕ 3 Λ + )] 3 = 1 τ ϕ + N 1π ( ϕ ) 3/ + 1 Λ ( ϕ ) ( 3/ λ + g 4 3N ) ϕ 4 ( ) λ g 4 3N ϕ 4 π 4! π Λ 4! with : τ = m g N Λ π and λ g 4 = Λ λ g 4 Minima : ϕ min = 4 π τ with τ < 0 Maxima : ϕ max = 3N π a Λ with a = 3N π λ g 4 5

53 Minima : ϕ min = 4 π τ with τ < 0 Maxima : ϕ max = 3N π a Λ with a = 3N π λ g V eff φ Lesson : As long as we keep only relevant operators, the potential is convex (continuum line) and we do not experience any bending. If we include an irrelevnt operator as ϕ 4, a maximum is formed at a scale O(Λ) and the potential bends down. But this happens beyond the scaling region! 53