First steps toward an asymptotically safe model of electroweak interactions

Transcription

1 First steps toward an asymptotically safe model of electroweak interactions (with M. Fabbrichesi, R. Percacci, F. Bazzocchi, L. Vecchi and O. Zanusso) Alberto Tonero SISSA Trieste - Asymptotic Safety Seminar June 6, / 28

2 Motivations The minimal realization of the SM would include only fundamental particles already discovered (no Higgs boson). EW symmetry implemented by coupling fermions and gauge bosons to the corresponding Nambu-Goldstone bosons described by a Nonlinear σ Model. Two strong arguments against this picture: violation of unitarity; EW precision measurements; Both these arguments may be avoided if this EW symmetry breaking sector is Asymptotically Safe. 2 / 28

3 The Nonlinear σ Model (NLσM) The NLσM action describes the physics of the Goldstone bosons (Gb) of a spontaneously broken symmetry (G H): S = 1 2f 2 d d x h αβ µ ϕ α µ ϕ β f is the Gb coupling with mass dimension (2 d)/2. 1 the flow of h f 2 αβ is governed by the Ricci tensor: d ( 1 f 2 h αβ ) = 2c d k d 2 R αβ. one-loop UV non-trivial fixed point (d > 2): f 2 = d 2 D 2 c d R. ( f 2 = k d 2 f 2 ) (R Ricci scalar) [A. Codello and R. Percacci, PLB 672 (2009) 280] 3 / 28

4 Gauging the NLσM Gauged SU(N) chiral NLσM Euclidean action (the left part of the isometry group SU(N) L SU(N) R is gauged): S = 1 2f 2 d d x h αβ D µ ϕ α D µ ϕ β + 1 4g 2 d d x FµνF i µν i g is the gauge coupling. D µ ϕ α = µ ϕ α + A i µri α (ϕ) is the gauge covariant derivative, Ri α-right invariant Killing vectors (α, i = 1,.., N 2 1). F i µν = µ A i ν ν A i µ + f i jla j µa l ν is the gauge field strength. the action is invariant under local SU(N) L infinitesimal transformations δ ɛ ϕ α = ɛ i LR α i (ϕ) δ ɛ A i µ = µ ɛ i L + f jl i A j µɛ l L. 4 / 28

5 Background field expansion and gauge fixing Expand the fieds around nonconstant backgrounds ϕ and Ā: ϕ α = ϕ α + ξ α 1 2 Γ β α γξ β ξ γ +... A i µ = Āi µ + a i µ. ξ α normal coordinates (preserve background invariance). Functional Taylor series expansion of the action: S(ϕ, A) = S( ϕ, Ā) + S(1) ( ϕ, Ā; ξ, a) + S(2) ( ϕ, Ā; ξ, a) +... The second order piece is: S (2) = 1 2f 2 d d x ξ α ( D 2 h αβ D µ ϕ ɛ D µ ϕ η ) R ɛαηβ ξ β + 1 2g 2 d d x a i µ + 1 f 2 d d x a i ( µ hαγ D µ ϕ α β R γ i + h αβri α D µ) ξ β, ( D 2 δ ij δ µν + D ν Dµ δ ij + F lµν f lij + g2 f 2 δ ijδ µν ) a j ν where D µ ξ α = µ ξ α + Āi µ β R α i ξβ. 5 / 28

6 Background gauge fixing term: S gf = 1 2αg 2 d d x δ ij χ i χ j with χ i = D µ a i µ + β g2 f 2 Ri αξ α The second order piece becomes: S (2) = 1 ( 2f 2 d d x ξ α D 2 h αβ D µ ϕ ɛ D µ ϕ η R ɛαηβ + β2 g 2 ) α f 2 h αβ ξ β + 1 ( ( 2g 2 d d x a i µ D 2 δ µν ij ) )δ ij D µ D ν 2F lµν f ilj + g2 α f 2 δµν ij a j ν f 2 d d x a µi D µ ϕ α h αγ β R γ i ξβ + 1 ( ) β f 2 α 1 d d x D µ a i µδ ij R j β ξβ + S gh. where S gh is the ghost action. The case α = β = 1 is the generalization of the t Hooft-Feynman gauge fixing to the background field method. 6 / 28

7 Functional RG study RG study is done by using ERGE and heat-kernel techniques: Γ k = dγ k = 1 ( δ 2 ) 1 ( 2 Tr Γ k δ 2 ) 1 δθδθ + Rθ k Ṙθk Tr Γ k δ cδc + Rc k Ṙck where θ T = (ξ i, a i µ) and t = log(k/k 0 ). Optimized cutoff kernels R θ k and Rc k : R θ k = ( 1 f 2 R k ( D 2 ξ ) g 2 R k ( D 2 a) ) R c k = R k( D 2 c ) The t-derivative of the cutoff is dr θ k = ( [ ] 1 f 2 t R k ( Dξ 2) + η ξr k ( Dξ 2) 0 [ 1 0 g 2 t R k ( Da) 2 + η a R k ( Da) ] 2 where η ξ = 2 t log f and η a = 2 t log g. 7 / 28 )

8 β-functions in d = 4 System of linear equations for the β functions: d 1 g 2 = N 8 ( (4π) η ) a 1 ( 9 (1 + g2 f 2 ) η a + 1 ) 1 8 η ξ 1 + g2 f 2 d 1 f 2 = N k 2 (4π) (1 + g2 f 2 )2 1 + η ξ One loop flow equations for g 2 f 2 : g f g2 d g2 = N 29 (4π) 2 4 g4 ( d f 2 = 2 1 3N ) 4(4π) 2 g2 f 2 1 N (4π) 2 4 f 4 f 2 ( 2 + η ) ξ + η a. 6 8 / 28

9 Fixed point - numerical result Table: Figure: cutoff and gauge f g type I, α = 1 4π 6/N 0 type II, α = 1 8π 2/3N 0 type II, α = 0 8π 2/3N t Figure: Running of f and g for N = 2 in d = 4, f / 28

10 SU(2) U(1) gauged NLσM If no Higgs, the spontaneous breaking of SU(2) U(1) U(1) would be implemented by coupling the gauge bosons to the NLσM Nambu-Goldstone bosons: S = 1 2f g 2 d 4 x h αβ D µ φ α D µ φ β + 1 4g 2 d 4 x B µν B µν d 4 x W i µνw µν i (Euclidean action) 1/f 2 = υ 2 /4 (υ is the EW VEV), g and g are the gauge couplings. the gauge covariant derivative is D µ φ α = µ φ α + W i µr α i B µ L α 3 α, i = 1, 2, 3. R/L are right/left invariant Killing vectors. 10 / 28

11 β-functions The system of RGEs for f 2, g 2 and g 2 is (at one-loop): d f { 2 = 2 f f 4 (4π) 2 4 (1 + m f 4 2 2g2 f W )2 4 (1 + m 2 + Z )2 (1 + m 2 W )3 [ ] 2 2 g f 1 + (1 + m 2 W )(1 + m2 B ) (1 + m 2 W ) + 1 (1 + m 2 B ) g 2 + f 2 [ ] } 1 (1 + m 2 W )(1 + m2 Z ) (1 + m 2 W ) + 1 (1 + m 2 Z ) dg 2 [ = g (4π) m 2 W (1 + m ] W )2 2 dg 2 = 1 g (4π) m 2 W where m 2 W = m2 W /k2 = g 2 / f 2, m 2 B = g 2 / f 2 and m 2 Z = (g2 + g 2 )/ f / 28

12 U.V. limit (g 2, g 2 f 2 ) of the RGEs: d f 2 dg 2 = 2 f 2 1 f 2 ( f 2 2 (4π) 2 + 6g 2 + 3g 2) = g4 29 (4π) 2 2 dg 2 g 4 = 1 6 (4π) 2 Approximation of constant gauge couplings (g = g = 0.65, g = g = 0.35): Approximate solution for f 2 (k): f = 64π 2 6g 2 3g f 2 (k) = f 2 f 2 0 f 2 + (k 2 k 2 0 )f / 28

13 Scattering amplitude and perturbative unitarity Elastic pion-pion scattering amplitude π i π j π k π l : A(π i π j π k π l ) = A(s, t, u)δ ij δ kl +A(t, s, u)δ ik δ jl +A(u, s, t)δ il δ jk where A(s, t, u) = sf 2 /4. Isospin amplitudes: A 0 = 3A(s, t, u) + A(t, s, u) + A(u, s, t) A 1 = A(t, s, u) A(u, s, t) A 2 = A(t, s, u) + A(u, s, t) Partial wave decomposition of A I (I = 0, 1, 2): t IJ = 1 64π 1 1 dcosθ P J (cosθ) A I The unitarity bound on the first partial wave: t 00 = sf 2 64π < / 28

14 RG improved unitarity bound f 2 64 Π f 2 32 Π f 2 16 Π f 2 8 Π 2 The position of the fixed point is regularization scheme dependent, inclusion of higher derivative operators may also move the fixed point [R. Percacci and O. Zanusso, Phys. Rev. D 81 (2010) ]. 14 / 28

15 S and T parameters L (0) and L (1) operators that give contribution to electroweak oblique parameters: L = 1 2f 2 h αβd µ ϕ α D µ ϕ β + 1 4g 2 W i µνw µν i + 1 4g 2 B µνb µν a 0 f 2 D µϕ α D µ ϕ β L 3 αl 3 β a 1 2 Bµν W i µνr iα L α 3 S and T parameters computed in our model: S = 16πa 1 (m Z ) + 1 [ ( )] 5 6π 12 log mh, m Z [ T = a 0 (m Z ) α em 8π cos 2 θ W 12 log ( mh m Z )]. 15 / 28

16 β functions I (ungauged case) Lagrangian in absence of gauge interactions: L = 1 2f 2 ĥαβ µ ϕ α µ ϕ β. New metric ĥαβ = L 1 αl 1 β + L2 αl 2 β + (1 2a 0)L 3 αl 3 β. RG flow of the NLσM: ( ) d 1 f 2 ĥαβ = 1 (4π) 2 k2 ˆRαβ In the basis of the right-invariant vectorfields: ˆR 11 = ˆR 22 = a 0, ˆR33 = 1 2 a 0. One-loop beta functions of f 2 and a 0 : d f 2 da 0 = 2 f 2 1 (4π) 2 f 4 ( ) a 0 = (4π) f 2 2 a 0 (1 2a 0 ). 16 / 28

17 Two nontrivial Fixed Points: FPI: f = 8π a 0 = 0 SU(2) R symmetric FPII: f = 4 2π a 0 = 1/2 SU(2) R broken f a 0 17 / 28

18 β functions II (gauged case) One-loop beta functions (g 2, g 2 f 2 ): d f 2 da 0 da 1 = 2 f 2 1 f 2 ( f 2 2 (4π) 2 (1 + 2a 0 ) + 6g 2 + 3g 2), = 1 ( 1 f 2 2 (4π) 2 a 0 (1 2a 0 ) + 32 ) g 2, ( 1 = f 2 (4π) 2 a ). 6 The two nontrivial Fixed Points of the ungauged case are slightly shifted: FPI: f = 25.1 a 0 = a 1 = (1 relevant and 2 irrelevant directions) FPII: f = 17.7 a 0 = a 1 = (2 relevant and 1 irrelevant directions) 18 / 28

19 Comparison with experimental bounds T S Figure: The half-line (FPII endpoints) and the dot (FPI endpoint) show the values permitted by asymptotic safety. The ellipses show the 1 and 2 σ experimental bounds with m H =117GeV [PDG, J. Phys. G, 37, (2010)]. 19 / 28

20 Fermions and Goldstone bosons SU(N) L SU(N) R invariant nonlinear sigma model lagrangian coupled to fermions: L = 1 ( ) f 2 Tr U µ UU µ U + ψ L iγ µ µ ψ L + ψ R iγ µ µ ψ R 2h f ( ψia L U ij ψ ja R + h.c.). (1/f = υ/2) U = e ifπa T a is SU(N) valued scalar field, π a Goldstone bosons. ψl/r ia in the fundamental of SU(N) L/R and SU(N c ) Degenerate fermion multiplet of mass h is the Yukawa coupling. m = 2 h f = hυ, 20 / 28

21 Beta functions One-loop RG equations for f and h using sharp cutoff regularization: d f dh = f N 64π f N c 4π 2 h2 f, ( 1 = 16π 2 4N c 2 N 2 ) 1 h N 2 2 N 64π 2 N h f 2. Fixed Points: FPI (h = 0, f = 0) trivial FPII (h = 0, f = 8π/ N) h = 0 at all scales FPIII (h 0, f 0) N > 2N c (not true for the most phenomenologically important case N = 2, N c = 3) 21 / 28

22 Four-fermion interactions Fix N = 2, we add to the lagrangian a complete set of SU(2) L SU(2) R four fermion operators: ( ) ( ) L ψ 4 = λ ψia 1 L ψ ja jb R ψ R ψib L + λ ψia 2 L ψ jb jb R ψ R ψia L ( + λ ψia 3 L γ µ ψl ia jb ψ L γµ ψ jb ia L + ψ R γ µ ψr ia jb ψ R γµ ψ jb R ( + λ ψia 4 L γ µ ψl ib jb ψ L γµ ψ ja ia L + ψ R γ µ ψr ib jb ψ R γµ ψ ja R ) ). These four-fermion interactions have been studied also by Gies, Jaeckel and Wetterich [PRD (2004)]; We do not seek to model chiral symmetry breaking; In our RG analysis we consider only the third family of quarks, ψ t = (t b); In the case of SU(2) U(1) there would be 10 operator. 22 / 28

23 Beta functions One-loop RG equations for f, h and λ i = k 2 λ i (sharp cutoff): d f dh d λ 1 d λ 2 d λ 3 d λ 4 = f 1 32π f N c 4π 2 h2 f [ 1 = 16π 2 4N c ] f (N c λ 1 + λ 2 ) 2 1 [ + f 2 64π 2 16(N c λ1 + λ ] 2 ) h = 2 λ 1 1 4π 2 = 2 λ π 2 = 2 λ π 2 = 2 λ π 2 h 3 [N c λ λ 1 λ2 2 λ 1 λ3 4 λ 1 λ4 ] [ 1 4 λ λ 1 λ3 + 2 λ 1 λ4 3 ] 4 λ (2N c 1) λ 2 λ3 [ 1 4 λ 1 λ2 + N ] c 8 λ (2N c 1) λ (N c + 2) λ 3 λ4 [ ] 1 8 λ λ 3 λ4 + (N c + 2) λ / 28

24 Fixed points table ( f = 17.78, h = 0) λ 1 λ2 λ3 λ4 ɛ h fp fp1a fp1b fp1c fp1d fp2a fp2b fp2c fp2d fp2e fp2f fp3a fp3b fp3c fp3d fp / 28

25 Numerical solution, initial conditions h 0 = m t /υ and f 0 = 2: 20 f, 10 h h f, no Λ i Running of f and h for N = 2 and N c = 3. Without four-fermion interactions, the AS behavior of f is destabilized around t = 3.5 ( 8, 3 TeV). 25 / 28

26 Experimental constraints Current bounds on contact interactions have been published for the case in which only one operator is considered [ E. Eichten, K. D. Lane, M. E. Peskin, PRL 50 (1983) 811], here ψ t = (u d): L qqqq = 4πA 2Λ 2 ia ψ L γ µ ψl ia jb ψ L γµ ψ jb L (A = ±1). The experimental bound is a lower bound of the so-called contact interaction scale Λ: λ(k) = 2π Λ 2. Current published bound [ATLAS Collaboration, arxiv: hep-ex]: Λ > 9.5TeV with 36 pb 1. Future expected bound [ATLAS and CMS, arxiv: hep-ph]: Λ > 30TeV with 100 fb / 28

27 We enforce the same bound on all the coefficients (conservative but unrealistic), λ i (k) < 2πk 2 /Λ bound : Λi Λ Figure: RG evolution of λ i towards the IR for the point fp1c. 27 / 28

28 Summary and Conclusions There seem to exist fixed point for the gauged NLσM in which only the leading two-derivative operator is considered. The position of the fixed point depends on the scheme of regularization and inclusion of higher derivative operators may also move the fixed point. In the case of the electroweak chiral lagrangian we were able to study some phenomenology emerging from the AS picture. AS seems to be compatible with electroweak precision measurement. It is possible to obtain estimations of S and T parameters in agreement with experimental data. Coupling the NLσM to fermion we have that, in the case of SU(2) U(1), the model is no more AS. AS can be restored introducing effective four-fermion interactions that satisfy current LHC experimental bounds on contact interactions. 28 / 28