Dilaton: Saving Conformal Symmetry

Transcription

1 Dilaton: Saving Conformal Symmetry Alexander Monin Ecole Polytechnique Fédérale de Lausanne December 2, 2013 lexander Monin (Ecole Polytechnique Fédérale de Dilaton: Lausanne) Saving Conformal Symmetry December 5, / 44

2 d 4 x ĝ( ˆΦ) 2 = d 4 x { g ( Φ) 2 + Φ 2 [ ( τ) ]} τ, e τ 2 e τ = ( τ) τ (1) [ a δ d n x ] g E 4 d 4 = a d n x g (E 4 + (d 4)σE 4 ), E 4 = R µνλσ R µνλσ 4Rµν R µν + R 2 (2) d 4 e iγ[j Φ,J X,gµν ] = (DδΦ) (DδX)e is[φ+δφ,x+δx,gµν ]+i gj Φ δφ+i gj X δx p 1 p 3 p 1 p 3 p 1 p 3 p 1 p 3 p 2 p 4 p 2 p 4 p 2 p 4 p 2 p 4 q 1 q 1 q 2 q 1 q 2 p 3 p 1 p 3 p 1 p 3 p 1 p 2 p 4 p 2 p 4 p 2 p 4 q 1 q 1 q 2 q 1 q 2 (a) (b) (c) (d) Γ[Φ, X, gµν ] = W [J Φ, J X, gµν ] d n x gφj Φ d n x gxj X, X (x ) = Ω X(x), ˆΦ (x ) = ˆΦ(x), [ a δ d n x ] g E 4 d 4 T µ µ = (4 d) λ 0 φ4 0 4! ĝ λσ (x) xλ x σ x µ x ν = ηµν e2τ(x) Ω 2 (x) X (x ) = Ω X(x) (3) = a d n x g (E 4 + (d 4)σE 4 ) L SB = e 2τ [( ˆΦ) 2 + ( τ) 2 m 2 ˆΦ 2 e 2τ ] d 4 2 = β(λ) [φ4 ], Tµν T λσ = # [ 1 d 1 4! 2 d p 2 2, (4) (P µλ P µλ + P µλ P µλ ) P µλ P µλ ] Alexander Monin (Ecole Polytechnique Fédérale de Dilaton: Lausanne) Saving Conformal Symmetry December 5, / 44 (5)

3 Motivation Purely academic Anomalies and certain properties of QFT Dilaton as a spectator (background) field has already been used Naturalness? Alexander Monin (Ecole Polytechnique Fédérale de Dilaton: Lausanne) Saving Conformal Symmetry December 5, / 44

4 Plan Symmetries and conservation laws Spontaneous symmetry breaking and coset construction Anomalies Conformal symmetry at the quantum level? Toy model up to 3 loops General argument Conclusions Alexander Monin (Ecole Polytechnique Fédérale de Dilaton: Lausanne) Saving Conformal Symmetry December 5, / 44

5 Symmetries are ubiquitous lexander Monin (Ecole Polytechnique Fédérale de Dilaton: Lausanne) Saving Conformal Symmetry December 5, / 44

6 Are symmetries fundamental? Lepton number conservation LHHL Λ Symmetry breaking operator Purification by the RG flow Within the Wilsonian approach global symmetries can be understood as accidents lexander Monin (Ecole Polytechnique Fédérale de Dilaton: Lausanne) Saving Conformal Symmetry December 5, / 44

7 Symmetry conservation law Space-time translations µ T µν = 0 U(1)-phase rotation µ j µ = 0 Symmetries provide building blocks and selection rules A µ j µ, LHeR, V (H H), etc. lexander Monin (Ecole Polytechnique Fédérale de Dilaton: Lausanne) Saving Conformal Symmetry December 5, / 44

8 Conformal symmetry Group of transformations leaving Minkowski metric invariant up to an overall factor g µν(x) = Ω(x)η µν x µ = x µ + ξ µ µ ξ ν + ν ξ µ = 2 d η µν ξ Solution ξ µ = a µ + ω µν x }{{} ν Poincaré: P µ,m µν dilatation: D {}}{ αx µ + 2(bx)x µ b µ x 2. }{{} special conformal: K µ j µ = ξ ν T µ ν, µ j µ C = 0 T µ µ = 0 lexander Monin (Ecole Polytechnique Fédérale de Dilaton: Lausanne) Saving Conformal Symmetry December 5, / 44

9 CFTs Commutation relations for SO(2, d) [J µν, J λσ ] = i (J µσ η νλ + J νλ η µσ J νσ η µλ J µλ η νσ ) Conformal group puts severe restrictions on correlators Poincaré + dilatations fix two point functions φ (x)φ (y) = Special conformal transformations x µ y µ = allow to fix 3-point functions. 1 x y 2 x µ y µ (1 2bx + b2 x 2 )(1 2by + b 2 y 2 ) Not about that! lexander Monin (Ecole Polytechnique Fédérale de Dilaton: Lausanne) Saving Conformal Symmetry December 5, / 44

10 Symmetry breaking Massive states exist! Phenomenological viability implies symmetry breaking Spontaneous symmetry breaking implies existence of Goldstone modes Promoting all dimensionful couplings to a dynamical field X realizes the scale symmetry nonlinearly m 2 Φ 2 λ ΦX Φ 2 X 2 It is enough to introduce only one additional degree of freedom the dilaton to realize nonlinearly the conformal symmetry lexander Monin (Ecole Polytechnique Fédérale de Dilaton: Lausanne) Saving Conformal Symmetry December 5, / 44

11 Nonlinear realization: coset construction Group H can be promoted to G by adding Goldstone modes π A for a given symmetry breaking pattern G H Action of the group G is realized non-linearly on the coset G/H Ω The form transforms not covariantly but not covariant derivatives Ω(π) = e iπa T A g : Ω(π ) = gω(π)h 1 (g, π) Ω 1 µ Ω = µ π A T }{{} A + ωµ a t }{{} a broken unbroken Ω 1 dω Ω 1 dω + h 1 dh 1 µ π A T A h ( µ π A T A ) h 1 Lagrangian is an arbitrary H-invariant function of π! Alexander Monin (Ecole Polytechnique Fédérale de Dilaton: Lausanne) Saving Conformal Symmetry December 5, / 44

12 Space-time symmetries In this case The form Ω = e ixµ P µ }{{} e iπa T A unbroken Ω 1 µ Ω = eµ[ ν Pν + µ π A T }{{} A broken + ω a µ t a }{{} unbroken again provides building blocks: vierbein e ν µ, covariant derivative π It is rather usual that some derivatives can be put consistently to zero Certain modes are dependent: Inverse Higgs mechanism π A T A O = 0 ] Alexander Monin (Ecole Polytechnique Fédérale de Dilaton: Lausanne) Saving Conformal Symmetry December 5, / 44

13 Example: Point particle in d = Symmetry breaking pattern: boost and space translation are broken O(1, 1) R 1,1 R Ω = e i P 0t e ip1π e ikη Transformation properties under the boost g = e ika : gω = e ika e i P 0t+iP 1π(t) }{{ e ika } εi(η+a)k }{{} t =t cosh a+π(t) sinh a η (t )=η(t)+a π (t )=t sinh a+π(t) cosh a Ω 1 dω = i P 0 (cosh η π sinh η)dt + ip 1 ( sinh η + π cosh η) dt + i η Kdt }{{} can be set to zero Alexander Monin (Ecole Polytechnique Fédérale de Dilaton: Lausanne) Saving Conformal Symmetry December 5, / 44

14 Example: Point particle in d = Symmetry breaking pattern: boost and space translation are broken O(1, 1) R 1,1 R Ω = e i P 0t e ip1π e ikη Transformation properties under the boost g = e ika : gω = e ika e i P 0t+iP 1π(t) }{{ e ika } εi(η+a)k }{{} t =t cosh a+π(t) sinh a η (t )=η(t)+a π (t )=t sinh a+π(t) cosh a Ω 1 dω = i P 0 (cosh η π sinh η)dt + ip 1 ( sinh η + π cosh η) dt + i η Kdt }{{} tanh η= π Alexander Monin (Ecole Polytechnique Fédérale de Dilaton: Lausanne) Saving Conformal Symmetry December 5, / 44

15 Example: Point particle in d = Symmetry breaking pattern: boost and space translation are broken O(1, 1) R 1,1 R Ω = e i P 0t e ip1π e ikη Transformation properties under the boost g = e ika : gω = e ika e i P 0t+iP 1π(t) }{{ e ika } εi(η+a)k }{{} t =t cosh a+π(t) sinh a η (t )=η(t)+a π (t )=t sinh a+π(t) cosh a Ω 1 dω = i P 0 (cosh η π sinh η)dt + ip 1 ( sinh η + π cosh η) dt + i η Kdt }{{}}{{} tanh η= π π Alexander Monin (Ecole Polytechnique Fédérale de Dilaton: Lausanne) Saving Conformal Symmetry December 5, / 44

16 Example: Point particle in d = Symmetry breaking pattern: boost and space translation are broken O(1, 1) R 1,1 R Ω = e i P 0t e ip1π e ikη Transformation properties under the boost g = e ika : gω = e ika e i P 0t+iP 1π(t) }{{ e ika } εi(η+a)k }{{} t =t cosh a+π(t) sinh a η (t )=η(t)+a π (t )=t sinh a+π(t) cosh a Ω 1 dω = i P 0 (cosh η π sinh η) dt + ip 1 ( sinh η + π cosh η) dt + i η Kdt }{{}}{{}}{{} tanh η= π π i 1 π 2 dt Lagrangian for a relativistic point-like particle! lexander Monin (Ecole Polytechnique Fédérale de Dilaton: Lausanne) Saving Conformal Symmetry December 5, / 44

17 Nonlinear realization: conformal symmetry For each spontaneously broken symmetry there is a dynamical field Dilaton from the coset construction An element of the coset is parametrized in a standard way The fundamental form Ω 1 dω = idx µ P µ }{{} e τ vierbein Ω = e i P µx µ e ikµcµ (x) e idτ(x) +D ( µ τ 2c µ ) }{{} µτ +K ν ( µ c ν + 2c µ c ν δµc ν 2 )... }{{} µc ν Covariant derivative µ τ can be consistently put to zero 2c µ = µ τ τ = 2c µ x µ, c µ = const : [τ(x)d + c µ (x)k µ ] O = 0 lexander Monin (Ecole Polytechnique Fédérale de Dilaton: Lausanne) Saving Conformal Symmetry December 5, / 44

18 Free massive scalar field in d = 4: Example L = 1 2 ( ˆΦ) m2 ˆΦ2 Vierbein: e τ and 2c µ c ν δ ν µc 2 ( τ) 2 As a result the Lagrangian for the non-linear realization has the form L SB = e 2τ 2 [( ˆΦ) 2 + ( τ) 2 m 2 ˆΦ2 e 2τ ] X = ve τ, ˆΦ = Φe τ L SB = 1 2 ( Φ)2 Φ X µφ µ X + 1 ( ) 2 ( Φ)2 1 + Φ2 X 2 m2 2v 2 X2 Φ 2 One degree of freedom is enough! lexander Monin (Ecole Polytechnique Fédérale de Dilaton: Lausanne) Saving Conformal Symmetry December 5, / 44

19 Alternative way to introduce the dilaton A theory with the action S[ˆΦ] is coupled to a background metric ĝ µν S[ˆΦ, ĝ µν ] Dilaton appears as a conformal mode of the metric ĝ µν = e 2τ g µν, ˆΦ = e τ Φ S SB [Φ, τ] = S[ˆΦ, ĝ µν ], g µν = η µν Obviously the Lagrangian for Φ and τ is invariant under Φ Φe σ, g µν e 2σ g µν, τ τ σ ˆΦ ˆΦ, ĝ µν ĝ µν lexander Monin (Ecole Polytechnique Fédérale de Dilaton: Lausanne) Saving Conformal Symmetry December 5, / 44

20 Example Free massive scalar field in d = 4 L = 1 2 ( ˆΦ) m2 ˆΦ2 S[ˆΦ, ĝ µν ] = 1 2 d 4 x ] ĝ [( ˆΦ) 2 m 2 ˆΦ2 + v2 6 ˆR Introducing X = ve τ L SB = 1 2 ( Φ)2 Φ X µφ µ X + 1 ( ) 2 ( Φ)2 1 + Φ2 m2 X 2 2v 2 X2 Φ 2 The metric for the σ-model is flat conformal invariance is manifest lexander Monin (Ecole Polytechnique Fédérale de Dilaton: Lausanne) Saving Conformal Symmetry December 5, / 44

21 From classical to quantum Anomalies Sometimes regularization breaks a symmetry µ j µ 0 Examples Chiral anomaly Pauli-Villars breaks e iαγ 5 explicitly Dim-reg breaks chiral symmetry due to specific definition of γ5 Trace anomaly. Regularizations introduce scale explicitly Dim-reg: couplings are dimensionful in d 4 lexander Monin (Ecole Polytechnique Fédérale de Dilaton: Lausanne) Saving Conformal Symmetry December 5, / 44

22 Trace anomaly Classical improved energy momentum tensor for λφ 4 is traceless T µν = µ φ ν φ η µν L ξ( µ ν η µν 2 )φ 2 At the quantum level dimensionally regularized trace is Coupling T µ µ = (4 d) λ 0φ 4 0 4! After renormalization Operator T µ µ = (4 d) 4! φ 4 4! ( λ + β(λ) ) ( 1 2β(λ) 4 d (4 d)λ }{{} λ 0 ) [φ 4 ] } {{ } φ 4 0 = β(λ) [φ4 ] 4! Alexander Monin (Ecole Polytechnique Fédérale de Dilaton: Lausanne) Saving Conformal Symmetry December 5, / 44

23 Conformal symmetry at the quantum level Regularization is a source of an explicit symmetry breaking Introduce regularization consistent with the symmetry Non-linear realization gives a hint how to introduce appropriate regularization: All the scales are dilaton related! Example, dim-reg φ 4 theory ( φ) 2 + v 2 e 2τ ( τ) 2 λφ 4 }{{} Classical d=4 ( φ) 2 + v 2 e 2τ ( τ) 2 λv # e (4 d)τ φ 4 }{{} Dim-reg d 4 lexander Monin (Ecole Polytechnique Fédérale de Dilaton: Lausanne) Saving Conformal Symmetry December 5, / 44

24 Another source of anomaly Regularization is not the end of the story Counter terms might break the symmetry! Example: Weyl anomaly d = 4: Regularized generating functional W [g µν ] is Weyl invariant e iw [gµν] = Dφe i 2 d 2 d ( φ)2 +ξφ 2R, ξ= 1 4 d 1 The counter term with E 4 = R µνλσ R µνλσ 4R µν R µν + R 2 is not [ a δ d n x ] g µν e g E 2σ g µν 4 = a d n x g σe 4 d 4 One has to make sure that counterterms respect the symmetry! lexander Monin (Ecole Polytechnique Fédérale de Dilaton: Lausanne) Saving Conformal Symmetry December 5, / 44

25 Example: 1 loop Dimensionally regularized d = 4 2ε Lagrangian can be expanded around X = v + χ µ 2ε L 0 = 1 2 ( φ) ( X)2 λ 4! φ4 X 2α, α= 4 d d 2 {}}{ φ 4 X 2α = φ 4 v [1 2α + }{{} 2ε (1 ln + χ ) ( ( + 2ε 2 v }{{} ln 1 + χ ) ( + ln χ )) + O ( ε 3)] v v Not important at one loop two loops Leading divergence is 1/ε #loops Renormalization Scale/conformally invariant 1-loop counter term 3 λ 2 v 2α φ 4 16π 2 2ε 4! = λ2 3 φ 4 (4π) 2 2ε 4! X2α + finite Alexander Monin (Ecole Polytechnique Fédérale de Dilaton: Lausanne) Saving Conformal Symmetry December 5, / 44

26 Example: 2 loops Diagrams to be taken into account ε 2 ε 2 ε 1 Alexander Monin (Ecole Polytechnique Fédérale de Dilaton: Lausanne) Saving Conformal Symmetry December 5, / 44

27 Example: 2 loops Diagrams to be taken into account ε 2 ε 2 ε 1 ε Alexander Monin (Ecole Polytechnique Fédérale de Dilaton: Lausanne) Saving Conformal Symmetry December 5, / 44

28 Example: 2 loops Diagrams to be taken into account ε 2 ε 2 ε 1 ε Alexander Monin (Ecole Polytechnique Fédérale de Dilaton: Lausanne) Saving Conformal Symmetry December 5, / 44

29 Example: 2 loops Diagrams to be taken into account ε 2 ε 2 ε 1 ε lexander Monin (Ecole Polytechnique Fédérale de Dilaton: Lausanne) Saving Conformal Symmetry December 5, / 44

30 Example: 2 loops Diagrams to be taken into account ε 2 ε 2 ε 1 ε Alexander Monin (Ecole Polytechnique Fédérale de Dilaton: Lausanne) Saving Conformal Symmetry December 5, / 44

31 Example: 2 loops Diagrams to be taken into account ε 2 ε 2 ε 1 ε Divergent ε 1 Alexander Monin (Ecole Polytechnique Fédérale de Dilaton: Lausanne) Saving Conformal Symmetry December 5, / 44

32 Example: 2 loops Diagrams to be taken into account ε 2 ε 2 ε 1 ε Finite! Alexander Monin (Ecole Polytechnique Fédérale de Dilaton: Lausanne) Saving Conformal Symmetry December 5, / 44

33 Example: 2 loops Diagrams to be taken into account ε 2 ε 2 ε 1 ε ε Finite! Alexander Monin (Ecole Polytechnique Fédérale de Dilaton: Lausanne) Saving Conformal Symmetry December 5, / 44

34 Example: 2 loops Diagrams to be taken into account ε 2 ε 2 ε 1 Standard renormalization of φ 4 λ2 1 (4π) 4 24ε ( φ)2 Alexander Monin (Ecole Polytechnique Fédérale de Dilaton: Lausanne) Saving Conformal Symmetry December 5, / 44

35 Example: 2 loops Diagrams to be taken into account ε 2 ε 2 ε 1 }{{} Standard renormalization of φ 4 [ ] λ2 1 λ ε φ 4 (4π) 4 4 ε 2 4! X2α (4π) 4 24ε ( φ)2 [ ] λ ε φ 4 [ ( (4π) 4 4 ε 2 4! v2α 1 + 4ε ln 1 + χ )] + finite v }{{} φ 4 renormalization Alexander Monin (Ecole Polytechnique Fédérale de Dilaton: Lausanne) Saving Conformal Symmetry December 5, / 44

36 Example: 3 loops Graphs from φ 4 { ε 3 Divergent graphs { New graphs Alexander Monin (Ecole Polytechnique Fédérale de Dilaton: Lausanne) Saving Conformal Symmetry December 5, / 44

37 Example: 3 loops Power of divergence = }{{} ε 3 # of vertices with χ ε Divergence of a graph At most 2 vertices!! New structures appear due to the dilaton in the loop! C 6 = L CT = C 6 6! φ 6 X 2 X2α C 8 8! λ4 (4π) ε, C 8 = λ4 φ 8 X 4 X2α 9625 (4π) ε lexander Monin (Ecole Polytechnique Fédérale de Dilaton: Lausanne) Saving Conformal Symmetry December 5, / 44

38 6d example L = 1 2 ( φ) ( X)2 g φ 4 X 2α6, 4! }{{} X Leads to ε unsuppressed vertices 2α 6= (6 d) (d 2) I II III L CT = 1 (2!) 2 C 1 6 d C 1 = Counterterms are conformally invariant! φ 2 φ2 2 X X + 1 C 2 (3!) 2 6 d φ 3 φ3 2 X2 X C 3 (4!) 2 6 d φ 4 φ4 2 X3 X 3 g 2 12(4π) 3 C 2 = g2 6(4π) 3 C 3 = g2 3(4π) 3 lexander Monin (Ecole Polytechnique Fédérale de Dilaton: Lausanne) Saving Conformal Symmetry December 5, / 44

39 8d example Classically conformal invariant Lagrangian has the form One of the counter terms L = 1 2 ( φ) ( X)2 g 3! φ 3 X# X1/3 δ 2 3 = g d 6(4π) 4 3! { φ 2 X 2/3 2 (φx 1/3 ) }{{} φx 1/3 [ (φx 1/3 ) ] 2 } Only for relative minus the expression is conformally invariant Alexander Monin (Ecole Polytechnique Fédérale de Dilaton: Lausanne) Saving Conformal Symmetry December 5, / 44

40 General argument 1PI effective action Γ reg [Φ] is a Legendre transform of W reg [J Φ ] e iwreg[jφ] = DΦ e is[φ]+i J Φ Φ reg If a regulator is consistent with a symmetry δγ reg [Φ] = 1 ε δγ P [Φ] + δγ F [Φ] = 0 For internal symmetries δ does not depend on the # of dimensions δγ P [Φ] = δγ F [Φ] = 0 Example: Weyl symmetry Alexander Monin (Ecole Polytechnique Fédérale de Dilaton: Lausanne) Saving Conformal Symmetry December 5, / 44

41 General argument Conformal transformations depend explicitly on d! Regularized effective action with metric Γ[Φ, X, g µν ] is Weyl invariant: ˆΦ(x) = e τ(x) Φ(x), ĝ µν (x) = e 2τ(x) g µν (x), ˆX = v Γ[Φ, X, g µν ] = Γ[ˆΦ, v, ĝ µν ] = 1 ε Γ P [ˆΦ, v, ĝ µν ] + Γ F [ˆΦ, v, ĝ µν ] Assuming no Diff anomaly Γ P [ˆΦ (x ), v, ĝ }{{} µν(x ) }{{} ] = Γ P [ˆΦ, v, ĝ µν ] ˆΦ(x) ĝ λσ (x) xλ x x σ µ x ν Alexander Monin (Ecole Polytechnique Fédérale de Dilaton: Lausanne) Saving Conformal Symmetry December 5, / 44

42 Conformal symmetry For the flat metric ĝ µν (x) = η µν e 2τ(x) conformal transformations lead to x µ = Ω 1 (x) (x µ a µ x 2 ) }{{} 1 2(ax)+a 2 x 2 ĝ µν(x ) = η µν e 2τ(x) Ω 2 (x) It can be written as transformation of Φ and X keeping g µν = η µν fixed e 2τ (x ) η µν ĝ µν(x ) = η µν e 2τ(x) Ω 2 (x) Diff { e τ (x ) Φ (x ) ˆΦ }} (x ) = }{{} ˆΦ(x) { Weyl Φ(x)e τ(x) As a result we get the standard conformal transformations! Φ (x ) = Ω Φ(x) e 2τ (x ) = e 2τ(x) Ω 2 (x) Alexander Monin (Ecole Polytechnique Fédérale de Dilaton: Lausanne) Saving Conformal Symmetry December 5, / 44

43 Conclusions Non-linearly realized conformal symmetry can be promoted to the quantum level Dilaton is a source for all scales in the theory Relevance for naturalness? Unitarity? Alexander Monin (Ecole Polytechnique Fédérale de Dilaton: Lausanne) Saving Conformal Symmetry December 5, / 44

44 Thank you very much Alexander Monin (Ecole Polytechnique Fédérale de Dilaton: Lausanne) Saving Conformal Symmetry December 5, / 44