Asymptotically free nonabelian Higgs models. Holger Gies

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1 Asymptotically free nonabelian Higgs models Holger Gies Friedrich-Schiller-Universität Jena & Helmholtz-Institut Jena & Luca Zambelli, PRD 92, (2015) [arxiv: ], arxiv:16xx.yyyyy Prologue: & M.M. Scherer, S. Rechenberger, L. Zambelli, EPJC 73, 2652 (2013)[arXiv: ],... ERG 2016, Trieste, September 2016

2 If you got a problem, just put a scalar field

3 If you got a problem, just put a scalar field..., massive gauge bosons, symmetry breaking, CP problem inflation, dark matter, dark energy,...

4 Standard model Higgs sector Extremely successful: Price to be paid: naturalness? # of parameters? δm 2 Λ 2 UV completion? Triviality?

5 Standard model Higgs sector Extremely successful: accepted currencies: SUSY technicolor extra-dim s Price to be paid: naturalness? δm 2 Λ 2... # of parameters? UV completion? Triviality?

6 Standard model Higgs sector Extremely successful: accepted currencies: SUSY technicolor extra-dim s... Price to be paid: naturalness? # of parameters? δm 2 Λ 2 UV completion? Triviality? FRG offers: + gravity [ERG 16: Rahmede,Ohta,Eichhorn] asymptotic safety [ERG 16: Sannino,Bond,Litim,Buyukbese]

7 UV completion: triviality problem λφ 4 theory: perturbation theory predicts its own failure 10 (Landau 55) (Gell-Mann, Low 54) = b 0 ln Λ, b 0 = 3 λ R λ Λ m R 16π λ R and m R fixed: ( = Λ L m R exp ) 1 b 0λ R Landau pole singularity lim (Λ/m R ) : = λ R 0 Triviality lattice evidence: (Luescher,Weisz 88; Hasenfratz,Jansen,Lang,Neuhaus,Yoneyama 87; Wolff 11; Buividovich 11; Weisz,Wolff 12,...)

8 Triviality in nonabelian Higgs models? SU(N) action: [ 1 S = d 4 x 4 F µνf i iµν + (D µ φ) (D µ φ) m2 φ φ + λ 8 (φ φ) 2] D ab ν = ν δ ab i g W i ν(t i ) ab (too) naive argument: gauge coupling g is asymptotically free = UV theory may reduce to pure scalar sector = scalar triviality in λ

9 Triviality in nonabelian Higgs models? closer look at perturbation theory: e.g., SU(2): analytic solution: t g 2 = β g 2 = b 0 g 4 t λ = β λ = Aλ 2 B λg 2 + Cg 4 (e.g., Gross, Wilczek 73) (mass-independent reg scheme, deep Euclidean region) b 0 = 43 48π 2, A = 3 4π 2, B = 9 16π 2, C = 9 64π 2 λ(g 2 ) = g2 2A [ B + ( )] tanh ln g2 Λ 2b 0 g 2 B = b 0 B, = B 2 4AC

10 Triviality in nonabelian Higgs models? analytic solution: λ(g 2 ) = g2 2A [ B + ( )] tanh ln g2 Λ 2b 0 g 2 B = b 0 B, = B 2 4AC (e.g., Gross, Wilczek 73) = total asymptotic freedom λ g 2, if > 0

11 Triviality in nonabelian Higgs models? analytic solution: λ(g 2 ) = g2 2A [ B + ( )] tanh ln g2 Λ 2b 0 g 2 B = b 0 B, = B 2 4AC (e.g., Gross, Wilczek 73) = total asymptotic freedom λ g 2, if > 0 Λ 1.0 BUT: < 0 for all SU(N 2): = Landau pole g2 lattice studies: 1.0 (Lang et al. 81; Kuhnelt et al. 83; Jersak et al. 85; Maas 13 15)

12 Total asymptotic freedom B = b 0 B, = B 2 4AC can be made positive by adjusting N and adding fermions (Gross,Wilczek 73; Chang 74; Fradkin,Kalashnikov 75; Salam,Strathdee 78) (Callaway 88; Giudice et al. 14; Holdom,Ren,Zhang 14) good news: gauge-higgs locking λ g 2 implies m 2 H m 2 W/Z... reduction of couplings (Zimmermann 84) bad news: residual symmetry generically too large: SU(2) many possibilities but no ordering principle

13 Asymptotically free Higgs sectors for > 0 gauge-higgs locking: λ g phase diagram in (λ, g 2 ) plane Λ g 2

14 Asymptotically free Higgs sectors for > 0 flow of gauge-rescaled coupling: ξ := λ g 2, tξ = β ξ = g 2 [ Aξ 2 + Bξ + C ] 1.0 phase diagram in (ξ, g 2 ) plane quasi fixed points: β ξ = 0 at g 2 > 0 Ξ = ξ +, ξ g 2

15 Things to ponder general relevance/meaning of quasi fixed points β ξ = 0 at finite g 2 different viewpoint: field rescalings, e.g. λ φ 4 = ξ ( g φ) 4 = for ξ ξ and g 0: large amplitude fluctuations expected... higher dimensional operators? life beyond the deep Euclidean region?... quasi-fixed point potentials with a minimum?... quasi-conformal vev?

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17 Effective field theory inclusion of higher-dimensional operators, e.g. potential: U(φ) = 1 2 m2 (φ φ) λ(φ φ) λ 3 Λ 2 eff (φ φ) Λ eff : UV scale of effective field theory

18 Effective field theory inclusion of higher-dimensional operators, e.g. potential: U(φ) = 1 2 m2 (φ φ) λ(φ φ) λ 3 48 k 2 (φ φ) k: sliding scale in the ERG spirit (Wetterich 93) UV behavior visible for k

19 Effective field theory inclusion of higher-dimensional operators, e.g. potential: U(φ) = N p n=1 λ 2n n!k 2(n 2) ( φ φ 2 ) n with N p

20 Effective field theory inclusion of higher-dimensional operators, e.g. potential: U(φ) = N p n=1 λ 2n n!k 2(n 2) ( φ ) n φ or 2 N p n=2 ( λ 2n φ ) n φ n!k 2(n 2) 2 v2 2 with N p conventional studies: λ 1 and v can be ignored in deep Euclidean region asymptotic symmetry (Lee,Weisberger 74)

21 Effective field theory inclusion of higher-dimensional operators, e.g. potential: U(φ) = N p n=1 λ 2n n!k 2(n 2) ( φ ) n φ or 2 N p n=2 ( λ 2n φ ) n φ n!k 2(n 2) 2 v2 2 with N p conventional studies: λ 1 and v can be ignored in deep Euclidean region asymptotic symmetry structure of the RG flow: t λ n = β λn (g 2, λ 1, λ 2,..., λ n+1 ) = infinite tower of coupled ODE s (Lee,Weisberger 74)

22 Effective field theory structure of the RG flow: t λ n = β λn (g 2, λ 1, λ 2,..., λ n+1 ) infinite tower of coupled ODE s t λ 2 = β λ2 (g 2, λ 1, λ 2, λ 3 ) t λ 3 = β λ2 (g 2, λ 1, λ 2, λ 3, λ 4 )..

23 Effective field theory structure of the RG flow: t λ n = β λn (g 2, λ 1, λ 2,..., λ n+1 ) e.g., truncating at N p = 2 (in deep Euclidean region): t λ 2 = β λ2 (g 2, λ 1, λ 2, λ 3 ) t λ 3 = β λ2 (g 2, λ 1, λ 2, λ 3, λ 4 )..

24 Effective field theory structure of the RG flow: t λ n = β λn (g 2, λ 1, λ 2,..., λ n+1 ) e.g., truncating at N p = 2 (in deep Euclidean region): t λ 2 = β λ2 (g 2, λ 1, λ 2, λ 3 ) t λ 3 = β λ2 (g 2, λ 1, λ 2, λ 3, λ 4 ).. BUT: since t λ 3 0, setting λ 3 = 0 is as good/bad as any other choice: e.g., λ 3 = const. or λ 3 = f (g 2 )

25 Effective field theory simplest agnostic approximation N p = 2 (deep E): t λ 2 = Aλ 2 2 B λ 2 g 2 + Cg 4 Dλ 3, D SU(2) = 1 4π 2

26 Effective field theory simplest agnostic approximation N p = 2 (deep E): t λ 2 = Aλ 2 2 B λ 2 g 2 + Cg 4 Dλ 3, D SU(2) = 1 4π 2 = asymptotically free trajectory, if λ 3 = ζ g 4, ζ = const. integrated flow: λ 2 (g 2 ) = g2 2A [ = B 2 4AC, B + tanh ( 2b 0 ln g2 Λ g 2 C = C Dζ )] = one-parameter family of asymptotically free trajectories

27 Asymptotically free trajectory: simple approximation simplest agnostic approximation N p = 2 (deep E): t λ 2 = Aλ 2 2 B λ 2 g 2 + Cg 4 Dλ 3, D SU(2) = 1 4π phase diagram in (λ 2, g 2 ) plane Λ e.g., ζ = g 2

28 Asymptotically free trajectory: simple approximation gauge-rescaled coupling: ξ 2 := λ 2 g 2, tξ 2 = β ξ2 = g 2 [ Aξ Bξ 2 + C Dζ ] 1.0 phase diagram in (ξ 2, g 2 ) plane quasi fixed points: β ξ = 0 at g 2 > 0 = perturbation about ξ + is RG irrelevant Ξ g 2

29 Asymptotically free trajectory: higher-orders? same conclusion for any finite truncation t λ 2 = β λ2 (g 2, λ 2, λ 3 ) t λ 3 = β λ2 (g 2, λ 2, λ 3, λ 4 ).. t λ n = β λn (g 2, λ 2, λ 3,..., λ n+1 ) if λ n+1 = const. g n 1 = one free parameter ζ.

30 Existence of asymptotically free trajectories? Crucial questions: Does the series expansion λ n (φ φ) n sum up to a global fixed point potential : U (φ)? = ζ parameter boundary condition for U (φ) Is U (φ) polynomially bounded (uniform convergence)? (Bridle,Morris 16; Morris 98)... singularity count, quantization of fixed-point potentials Are perturbations about U (φ) self-similar? (Morris 98)... quantization of RG directions, predictivity Classification of perturbations?... # of physical parameters,...

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32 ... yet another parameter: P-scaling solutions general field rescalings φ g P φ ξ n = g 2Pn λ n 0.4 Λ 2 = quasi fixed points for all values of P > 0 e.g., P {0.1, 0.2, 0.3, 0.4}, ζ = g2 = ζ and P parametrize boundary conditions for correlation functions

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34 Functional RG analysis RG flow equation: t Γ k k k Γ k = 1 2 Tr tr k (Γ (2) k (Wetterich 93) + R k ) 1 = functional differential equation initial value problem in RG time t fixed point condition for effective potential: 2 nd order ODE: 2 initial/boundary conditions

35 Functional RG analysis RG flow of the dimensionless scalar potential, u = k d U(φ) } t u = du+(d 2+η φ ) ρu +2v d {(d 1) N 2 1 i=1 l (G)d 0 (µ 2 W,i ( ρ))+(2n 1)l(B)d 0 (u )+l (B)d 0 (u +2 ρu ) { η φ = 8v d d ρ(3u +2 ρu ) 2 m (B)d 2,2 (u +2 ρu,u +2 ρu )+(2N 1) ρu 2 m (B)d 2,2 (u,u ) 2g 2 (d 1) N N 2 1 a=1 i=1 T iˆnat i aˆn l(bg)d 1,1 (u,µ 2 W,i)+(d 1) N 2 1 i=1 µ 4 [ W,i ρ 2a1(µ d 2 W,i)+m (G)d 2 (µ 2 W,i) ]} ρ= ρ min 2nd order PDE in k and φ (HG,Scherer,Rechenberger,Zambelli 13)

36 Functional RG analysis RG flow of the dimensionless scalar potential, u = k d U(φ) } t u = du+(d 2+η φ ) ρu +2v d {(d 1) N 2 1 i=1 l (G)d 0 (µ 2 W,i ( ρ))+(2n 1)l(B)d 0 (u )+l (B)d 0 (u +2 ρu ) { η φ = 8v d d ρ(3u +2 ρu ) 2 m (B)d 2,2 (u +2 ρu,u +2 ρu )+(2N 1) ρu 2 m (B)d 2,2 (u,u ) 2g 2 (d 1) N N 2 1 a=1 i=1 T iˆnat i aˆn l(bg)d 1,1 (u,µ 2 W,i)+(d 1) N 2 1 i=1 µ 4 [ W,i ρ 2a1(µ d 2 W,i)+m (G)d 2 (µ 2 W,i) ]} ρ= ρ min 2nd order PDE in k and φ gauge-rescaled variables: (HG,Scherer,Rechenberger,Zambelli 13) (HG,Zambelli 15) x = g 2P Z φ φ 2 k 2, f (x) = u t f = β f 4 f + (2 + η φ Pη W )xf { π 2 3 N 2 1 i=1 l (G)4 0T (g 2(1 P) ω 2 W,i (x))+(2n 1)l(B)4 0 (g 2P f )+l (B)4 0 (g 2P (f +2xf )) }

37 Global fixed point solution I weak-coupling expansion e.g., for P = 1 and SU(2): ( ) 2 [ ( )] 3 x f (x) = ζx 2 2x + x 2 ln 16π 2 + x (HG,Zambelli 15) fixed-point solution Coleman-Weinberg type, one-parameter family ζ

38 Global fixed point solution I weak-coupling expansion e.g., for P = 1 and SU(2): ( ) 2 [ ( )] 3 x f (x) = ζx 2 2x + x 2 ln 16π 2 + x (HG,Zambelli 15) fixed-point solution Coleman-Weinberg type, one-parameter family ζ leading-order scaling solution for different P: ζx 2 ξ 3 16π g 2P x for P (0, 1/2) 2 f (x) = ζx 2 3(3+8ζ) 128π g x for P = 1/2 2 ζx π g 2(1 P) x for P (1/2, 1) 2.

39 Asymptotically free UV gauge scaling solutions gauge scaling towards flatness, P = 1 (Rechenberger,Scherer,HG,Zambelli 13; HG,Zambelli 15) g = g = 0.04 g = 0.03 g = 0.02 g = 0.01 approach to UV k : φ 2 g 2 0, φ min 2 1 g 2, λ g4 0, m 2 W k 2 const. = deep Euclidean region is sidestepped... no asymptotic symmetry

40 Asymptotically free perturbations classification of (ir-)relevant perturbations for given ζ and P: δm 2 δg 2 δf (x) relevant marginally relevant as usual naturalness? natural marginal-relevant direction: 10 6 f f 6 = self-similar & polynomially bounded as should be (Morris 98) x 4

41 Global fixed point solution II origin of the free parameter ζ: singularity count (Dietz,Morris 12; Demmel,Saueressig,Zanusso 15) e.g. scalar models: u = 1 ρ e(u, u ; ρ) s(u, u ; ρ), ρ = 1 2 φ 2 2 nd order PDE 1 fixed singularity at ρ = 0 1 movable singularity at s(u, u ; ρ) = 0 = = 0 = 0 parameter family of solutions: only discrete quantized solutions... e.g., Wilson-Fisher FP in d = 3

42 Global fixed point solution II origin of the free parameter ζ: singularity count nonabelian Higgs model: f = 1 x e(f, f ; x) s(f, f ; x) 2 nd order PDE 1 fixed singularity at ρ = 0 1 movable singularity at s(f, f ; x) = 0 = = 0?

43 Global fixed point solution II origin of the free parameter ζ: singularity count nonabelian Higgs model: f = e(f, f ; x) x s(f, f ; x) 2 nd order PDE 1 fixed singularity at ρ = 0 1 movable singularity at s(f, f ; x) = 0 = = 1 BUT: gauge loop induces log-like singularity at x = 0 = 1 parameter family of solutions: ζ with f and f regular at x = 0

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45 Estimates of IR Observables Higgs to W boson mass ratio: mh 2 4mW m 2 H m 2 W c P ζ UV-IR mapping of physical parameters: v m H m W δm 2 relevant δg 2 δf marginal-relevant ζ, P exactly marginal Ξ = pheno-relevant parameter regime is accessible

46 Conclusions Interplay: asymptotic freedom boundary conditions b.c. s for correlation functions Non-abelian Higgs models can be asymptotically free and UV complete if our choice of b.c. s is legitimate scaling solutions satisfy necessary criteria self-similarity, boundedness, predictivity

47 Conclusions Interplay: asymptotic freedom boundary conditions b.c. s for correlation functions Non-abelian Higgs models can be asymptotically free and UV complete scaling solutions satisfy necessary criteria if our choice of b.c. s is legitimate self-similarity, boundedness, predictivity