Gravitational Renormalization in Large Extra Dimensions

Transcription

1 Gravitational Renormalization in Large Extra Dimensions Humboldt Universität zu Berlin Institut für Physik October 1, 2009 D. Ebert, J. Plefka, and AR J. High Energy Phys (2009) 028. T. Schuster and AR Phys. Rev. D 79 (2009) arxiv: [hep-th].

2 Outline 1 Introduction 2 Gravitational One Loop Divergences Yang-Mills Theory Scalar Field Fermions

3 Contents 1 Introduction 2 Gravitational One Loop Divergences Yang-Mills Theory Scalar Field Fermions

4 Renormalization loop diagrams are in general UV divergent Regularization by momentum cut-off Λ or dimensional regularization d = 4 ɛ Renormalization by adding counterterms introduces renormalization scale µ comparison with bare theory yield running of couplings g(µ) and other parameters, e. g. masses, which is encoded in the β function β g µ dg dµ

5 Gravitation Gravity as an effective field theory Perturbatively quantized Einstein gravity is famously nonrenormalizable Nevertheless it can be treated consistently as effective field theory [Donoghue 95] S grav = d d x { g Λ + 1 } R + c 1 R 2 + c 2 R µν R µν πG N The higher derivative terms arise in loop expansions, the values of their couplings are largely undetermined, e.g. c 1, c [Stelle 78] Question in this framework: Does inclusion of gravity alter the running of the Standard Model couplings g Maxwell (µ), g YM (µ), Y Yukawa (µ), λ ϕ 4(µ),? [Robinson,Wilczek 06]

6 Gravitation Gravity as an effective field theory Perturbatively quantized Einstein gravity is famously nonrenormalizable Nevertheless it can be treated consistently as effective field theory [Donoghue 95] S grav = d d x { g Λ + 1 } R + c 1 R 2 + c 2 R µν R µν πG N The higher derivative terms arise in loop expansions, the values of their couplings are largely undetermined, e.g. c 1, c [Stelle 78] Question in this framework: Does inclusion of gravity alter the running of the Standard Model couplings g Maxwell (µ), g YM (µ), Y Yukawa (µ), λ ϕ 4(µ),? [Robinson,Wilczek 06]

7 Gravitation In four dimensions g [Robinson,Wilczek 06] in background field method and cut-off regularization: β g κ 2 = 1 16π E2 κ 2 g [Pietrykowski 06], [Toms 07] and [Ebert, Plefka, AR 07] found independently: β g κ 2 = log[10](e/gev)

8 Gravitation In four dimensions g [Robinson,Wilczek 06] in background field method and cut-off regularization: β g κ 2 = 1 16π E2 κ 2 g [Pietrykowski 06], [Toms 07] and [Ebert, Plefka, AR 07] found independently: β g κ 2 = log[10](e/gev)

9 Gravitation The Einstein-Hilbert Action S = d d x g 2 R + L κ 2 matter (g µν, Φ) + L gf + L gh dimensionful coupling κ = 32πG N, [κ] = 1 d 2 Splitting the metric in flat background and graviton field: with h h α α. g µν η µν + κh µν ) g = 1 + κ 2 h + κ2 8 (h 2 2h αβ h αβ + O(κ 3 ) g µν = η µν κh µν + κ 2 h µα h α ν + O(κ 3 ) R = κ ( h µ ν h µν ) + O(κ 2 )

10 Gravitation The Einstein-Hilbert Action S = d d x g 2 R + L κ 2 matter (g µν, Φ) + L gf + L gh dimensionful coupling κ = 32πG N, [κ] = 1 d 2 Splitting the metric in flat background and graviton field: g µν η µν + κh µν ) g = 1 + κ 2 h + κ2 8 (h 2 2h αβ h αβ + O(κ 3 ) g µν = η µν κh µν + κ 2 h µα h α ν + O(κ 3 ) R = κ ( h µ ν h µν ) + O(κ 2 ) with h h α α. We use de Donder gauge: ν h µν 1 2 µh = 0

11 Gravitation Expansion in the field variables yields Feynman rules for vertices of arbitrary order in h: h 0 h 1 h 2 h 3 h 4 A 2 A A 4... z. B. L O(h,A 2 ) = κ µa a ν [ρ A a ( σ] h µρ η νσ + η µρ h νσ 1 2 h ηµρ η νσ) Which vertices are needed?

12 Gravitation Expansion in the field variables yields Feynman rules for vertices of arbitrary order in h: h 0 h 1 h 2 h 3 h 4 A 2 A A 4... z. B. L O(h,A 2 ) = κ µa a ν [ρ A a ( σ] h µρ η νσ + η µρ h νσ 1 2 h ηµρ η νσ) Which vertices are needed?

13 Large Extra Dimensions Space Time topology M = R 1,3 T δ The coordinates are denoted by X M = (x µ, y i ) In large extra dimension senarios the actual gravity scale can be much lower as the experienced Planck scale on the brane [ADD 98]: M(d=4) 2 = } (2πR)δ {{} V δ M D 2 (D)

14 Large Extra Dimensions S = d D xl bulk + d d xl brane For a generic Bulk field Φ: d D x 1 2 MΦ M Φ = d d x n Z δ with Φ(X) = V 1/2 δ n Z δ Φ( n) (x)e i n y R The induced brane metric 1 2 µφ ( n) µ Φ ( n) m2 ( n) Φ2 ( n) and m 2 ( n) = n n R 2 G MN (X) = η MN +κ (D) h MN (X) = g µν (x) = η µν +κ (d) n Z δ h ( n) + brane tension effects µν (x)

15 Large Extra Dimensions S = d D xl bulk + d d xl brane For a generic Bulk field Φ: d D x 1 2 MΦ M Φ = d d x n Z δ with Φ(X) = V 1/2 δ n Z δ Φ( n) (x)e i n y R The induced brane metric 1 2 µφ ( n) µ Φ ( n) m2 ( n) Φ2 ( n) and m 2 ( n) = n n R 2 G MN (X) = η MN +κ (D) h MN (X) = g µν (x) = η µν +κ (d) n Z δ h ( n) + brane tension effects µν (x)

16 Large Extra Dimensions S = d D xl bulk + d d xl brane For a generic Bulk field Φ: d D x 1 2 MΦ M Φ = d d x n Z δ with Φ(X) = V 1/2 δ n Z δ Φ( n) (x)e i n y R The induced brane metric 1 2 µφ ( n) µ Φ ( n) m2 ( n) Φ2 ( n) and m 2 ( n) = n n R 2 G MN (X) = η MN +κ (D) h MN (X) = g µν (x) = η µν +κ (d) n Z δ h ( n) + brane tension effects µν (x)

17 Contents 1 Introduction 2 Gravitational One Loop Divergences Yang-Mills Theory Scalar Field Fermions

18 Yang-Mills Theory The Yang-Mills Lagrangian L YM = 1 2 gg µν g ρσ tr[f µρ F νσ ] F µν = µ A ν ν A µ ig[a µ, A ν ] F a µν = µ A a ν ν A a µ + gf abc A b µa c ν Due to Slavonov-Taylor indentities only the renormalization of the two-point function is needed to calculate the gravity contribution to β g The higher derivative (HD) terms expected to appear at one-loop are { tr {D µ F µρ D ν F νρ } and tr F β α F γ β } F γ α.

19 Yang-Mills Theory Graviton Loop Corrections to the Gluon Two and Three Point Function (a) (b) 1 2 κ 2 (c) (d) (e) 1 2 gκ 2

20 Yang-Mills Theory Graviton Loop Corrections to the Gluon Two and Three Point Function (a) (b) 1 2 κ 2 (c) (d) (e) 1 2 gκ 2

21 Yang-Mills Theory Graviton Loop Corrections to the Gluon Two and Three Point Function (a) (b) 1 2 κ 2 (c) (d) (e) 1 2 gκ 2

22 Yang-Mills Theory Graviton Loop Corrections to the Gluon Two and Three Point Function (a) (b) 1 2 κ 2 (c) (d) (e) 1 2 gκ 2

23 Yang-Mills Theory Graviton Loop Corrections to the Gluon Two and Three Point Function (a) (b) 1 2 κ 2

24 Yang-Mills Theory The Gluon Graviton Vertices with Two Gluon Lines α β µ a ν b p q ν b µ a p γδ q αβ = iκδ ab[ P µν,αβ p q + η µν p (α q β) ηαβ p ν q µ p ν η µ(α q β) q µ η ν(α p β)] = i 2 κ2 δ ab[ (p ν q µ p q η µν )P αβ,γδ +p q(2i µν,α(γ η δ)β + 2I µν,β(γ η δ)α I µν,αβ η γδ I µν,γδ η αβ ) +2p (α q β) P µν,γδ + 2p (γ q δ) P µν,αβ + { 2p α η ν[µ η β](γ q δ) + 2p γ η ν[µ η δ](α q β) p ν (q α P µβ,γδ + q β P αµ,γδ +q γ P αβ,µδ + q δ P αβ,γµ ) } + {(p, µ) (q, ν)} ] where we have defined P µν,αβ 1 2 (ηµα η νβ + η µβ η να η µν η αβ ).

25 Yang-Mills Theory Field-Strength Renormalization κ 2 = iδ ab (q 2 η µν q µ q ν ) + iδ ab q 2 (q 2 η µν q µ q ν ) + = iκ 2 d 4 8d ( d 2 4d (d 2)(d 8) (D 2) = iκ 2 (d 2) 2 ( 3 d 2(d + 2)(D 2) + δ d (4 3 2 d)) n ) n d d k (2π) d 1 k 2 m 2 n d d k 1 (2π) d k 2 (k 2 m n 2)

26 Yang-Mills Theory Field-Strength Renormalization κ 2 = iδ ab (q 2 η µν q µ q ν ) + iδ ab q 2 (q 2 η µν q µ q ν ) + = iκ 2 d 4 8d ( d 2 4d (d 2)(d 8) (D 2) ) n d d k 1 (2π) d k 2 m n 2 Gravitational contribution to the YM β function in 4+δ dimensions β g κ 2 = 0

27 Yang-Mills Theory Field-Strength Renormalization κ 2 = iδ ab (q 2 η µν q µ q ν ) + iδ ab q 2 (q 2 η µν q µ q ν ) + = 2 (4π) D/2 1 Γ( D 2 ) d 4 (D 2) 2 ( (d 3)(d 2) + δ d (d2 4d + 8) ) Λ D 2 µ D 2 M D 2 (D) Gravitational contribution to the YM β function in 4+δ dimensions β g κ 2 = 0

28 Yang-Mills Theory Graviton Loop Corrections to the Gluon Two and Three Point Function (c) (d) (e) 1 2 gκ 2

29 Yang-Mills Theory The Divergences of the Three Gluon Graphs κ 2 = g f abc[ η µν (p ρ (2p q + p k + 3q k) q ρ (2q p + q k + 3p k)) +... (k µ k ν (p q) ρ +... ) 3(p ρ q µ k ν p ν q ρ k µ ) ] In four (brane) dimensions the only counterterm needed in one-loop Einstein-Yang-Mills theory is L c.t. YM = 1 (4π) 1+δ/2 Γ ( 8 Λ δ µ δ δ 2 + 1) 3δ M δ+2 tr {D µ F µρ D ν F νρ } (4+δ)

30 Yang-Mills Theory The Divergences of the Three Gluon Graphs κ 2 = g f abc[ η µν (p ρ (2p q + p k + 3q k) q ρ (2q p + q k + 3p k)) +... (k µ k ν (p q) ρ +... ) 3(p ρ q µ k ν p ν q ρ k µ ) ] In four (brane) dimensions the only counterterm needed in one-loop Einstein-Yang-Mills theory is L c.t. YM = 1 (4π) 1+δ/2 Γ ( 8 Λ δ µ δ δ 2 + 1) 3δ M δ+2 tr {D µ F µρ D ν F νρ } (4+δ)

31 Scalar Field L s = gg µν [ (D µ φ) D ν φ m 2 φ φ φ + λ 2 (φ φ) 2 ] D µ = µ iga µ The higher derivative terms expected to appear at one-loop are (D µ φ) D µ φ, (D 2 φ) D 2 φ, ig(d µ φ) F µν D ν φ, g 2 φ F µν F µν φ.

32 Scalar Field (D µ φ) D µ φ, (D 2 φ) D 2 φ, ig(d µ φ) F µν D ν φ, g 2 φ F µν F µν φ

33 Scalar Field (D µ φ) D µ φ, (D 2 φ) D 2 φ, ig(d µ φ) F µν D ν φ, g 2 φ F µν F µν φ

34 Scalar Field + crossed + crossed + crossed + crossed + crossed + crossed + crossed + crossed + crossed + crossed + crossed + crossed (D µ φ) D µ φ, (D 2 φ) D 2 φ, ig(d µ φ) F µν D ν φ, g 2 φ F µν F µν φ

35 Scalar Field Scalar field-strength and HD renormalization m φ = 0 L c.t. s = [ i 2(8 + 5δ) (4π) 1+δ/2 Γ ( δ 2 + 2) (δ + 2) 2 (D µφ) D µ φ Λδ+2 µ δ+2 M δ+2 { (D 2 φ) D 2 φ 1 3 ig(d µφ) F µν D ν φ g2 φ F µν F µν φ (4+δ) } Λ δ µ δ ] M δ+2 (4+δ) Note: Without extra dimension (δ = 0) the HD corrections vanish.

36 Scalar Field Renormalization of the ϕ 4 coupling for a real scalar L r = Z ϕ 1 2 ϕ 2 ϕ Z m 2 ϕ 1 2 m2 ϕϕ 2 Z ϕ 4 λ 4! ϕ Z ϕ 1 = Z m 2 ϕ 1 = Z ϕ 4 1 = κ2 2 16π 2 m2 ϕ d 4 κ2 2 16π 2 m2 ϕ d 4 κ2 2 16π 2 4m2 ϕ d 4 β Function β λ κ 2 = κ2 4π 2 m2 ϕλ

37 Fermions L f = g ψ(i /D m ψ Yϕ)ψ /D = γ a e µ a D µ D µ = µ iω µ iga µ The higher derivative terms expected to appear at one-loop are iψ /Dψ, iψ /D /D /Dψ, iψ /DD 2 ψ, iψd 2 /Dψ, iψd µ /DD µ ψ.

38 Fermions iψ /Dψ, iψ /D /D /Dψ, iψ /DD 2 ψ, iψd 2 /Dψ, iψd µ /DD µ ψ

39 Fermions Fermionic field-strength and HD renormalization m ψ = 0 L c.t. f = i (4π) 1+δ/2 Γ ( δ 2 + 2) ψ { + ( [ 3(11 + 9δ) (δ + 2) 2 Λ δ+2 µ δ+2 M δ+2 (4+δ) 8 δ) /D /D /D ( δ) /DD2 /D+ + ( δ)d2 /D + ( δ)d µ /DD µ} Λ δ µ δ δ M δ+2 (4+δ) ] ψ

40 Fermions Renormalization of the Yukawa coupling for uncharged fermion and scalar L r = Z ψ ψi/ ψ Z mψ m ψ ψψ Z ϕψψ Y ϕψψ +...

41 Fermions Z ψ 1 = κ π 2 4 m2 ψ d 4 Z mψ 1 = κ π 2 4 m2 ψ d 4 Z ϕψψ 1 = κ2 16π 2 ( 3 4 m2 ψ m2 ϕ) d 4 β Function β Y κ 2 = κ2 16π 2 (m2 ψ 1 2 m2 ϕ)y

42 Summary and Outlook The Yang-Mills β-function receives no contributions form gravitational self-coupling. Yukawa and ϕ 4 interactions receive gravitational corrections The spectrum of HD corrections is not restricted to Lee-Wick terms. Next step: Non-Perturbative Methods Functional Renormalization Group Equation µ µ Γ µ[φ] = 1 [ 2 Tr µ R ( ) ] µ Γ (2) µ [Φ]R µ µ [Wetterich 93,Reuter 98]

43 Thank you!