A manifestly scale-invariant regularization and quantum effective operators

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Ζοροβάβελ Τομαραίοι
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1 A manifestly scale-invariant regularization and quantum effective operators D. Ghilencea Corfu Summer Institute, Greece - 8 Sep 2015 E-print: arxiv: sponsored by RRC grant - project PN-II-ID-PCE

2 1 ] Outline - Introduction: Scale invariance as a solution to the hierarchy problem - Problem: Usual regularizations of quantum corrections break scale invariance - Goal: Study implications of a special, scale-invariant regularization. - Implications: New corrections to scalar potential beyond Coleman-Weinberg. - Applications: the scalar potential in SM + dilaton.

3 2 ] Introduction - One approach to hierarchy problem: scale invariance (x ρx, φ ρ d φ): forbids (higgs) mass terms - the real world is not scale invariant this symmetry must be broken. - at classical level: one can start with a scale invariant L - at quantum level? the need for a subtraction/renormalization scale (µ) Cutoff schemes: lnλ/m Z = lnλ/µ+lnµ/m Z, (Λ ). DR scheme: λ φ = µ 2ǫ λ (r) φ + n a n/ǫ n ], (ǫ 0) At quantum level: scale symmetry is broken explicitly by: a dimensionful scale (cutoff, Pauli Villars) or a dimensionful coupling (DR scheme).

4 3 ] Problem: in theories with scale/conformal symmetry: regularization (DR, etc...) breaks explicitly the very symmetry one wants to study at quantum level! - impact, particulary in non-renormalizable case, and for the hierarchy problem - usual (naive?) argument: DR breaks scale symmetry more softly ( ) Bardeen 1995] Solution: replace µ f(dilaton: σ). Deser 1970, Englert 1976, Shaposhnikov 2009] Evanescentpowerµ 2ǫ inthelastequation need σ 0 spontaneousbreakingofscaleinvariance! Goal: study its implications. ( ) if the DR breaking of scale symmetry were indeed soft the result should be similar to spontaneous breaking of scale symmetry - see later

5 4 ] Scale invariance at classical level L of two real scalar fields: An example: L = 1 2 µφ µ φ+ 1 2 µσ µ σ V(φ,σ) V = λ φ 4 φ4 + λ m 2 φ2 σ 2 + λ σ 4 σ4 Extremum: φ λ φ φ 2 +λ m σ 2] = 0, σ λ m φ 2 +λ σ σ 2] = 0, a) The ground state is σ = 0, φ = 0 and both fields are massless. b) IF σ 0 a solution, then φ 0; a non-trivial ground state exists if λ 2 m = λ φ λ σ ; λ m < 0. φ 2 σ = λ m, V = λ ( φ φ 2 + λ ) 2 m σ 2 2 λ φ 4 λ φ Spontaneous breaking of scale symmetry EWSB at tree-level, with a vanishing cosmo constant

6 5 ] Scale invariance at classical level L of two real scalar fields: L = 1 2 µφ µ φ+ 1 2 µσ µ σ V(φ,σ) An example: Kobakhidze et al 2007, 2014] V = λ φ 4 φ4 + λ m 2 φ2 σ 2 + λ σ 4 σ4 The mass eigenstates: m 2 φ = 2λ φ ( 1 λm /λ φ ) φ 2 = 2λ m (1 λ m /λ φ ) σ 2 m σ = 0 σ: Goldstone mode of scale invariance (dilaton). Shaposhnikov et al 2009, Ross et al 2014] Expect: σ M Planck To ensure a hierachy m φ φ O(100) GeV, one tunes classically λ m : φ σ if λ φ λ m λ σ, λ 2 m = λ φ λ σ λ m 1/ σ 2, λ σ 1/ σ 4. At quantum level: is extra tuning needed?

7 6 ] Scale invariance at quantum level L = 1 2 µφ µ φ+ 1 2 µσ µ σ V(φ,σ) - DR: d = 4 2ǫ: L] = d, λ]= V (4) ] =d 4(d 2)/2 = 4 d λ µ 4 d λ. A scale invariant regularization: µ µ(σ,φ). Then V Ṽ µ(φ,σ)4 d V U = Ṽ i d d p 2 (2π) Trln p 2 M 2 (φ,σ)+iε ], ( M 2 ) d αβ = 2 Ṽ α β Ṽαβ, = Ṽ 1 2 M 4 64π 2 s 4 d ln M ] s/κ 2, (M 2 ) αβ = V αβ ; α,β = φ,σ. s=φ,σ ( M 2 ) αβ = µ 4 d (M 2 ) αβ +(4 d)µ 2 N αβ ], ] M s 4 = µ TrM 2(4 d) 4 +2 (4 d)µ 2 Tr(M 2 N), s=φ,σ N αβ µ(µ α V β +µ β V α )+(µµ αβ µ α µ β )V, Evanescent corrections to (M 2 ) αβ bring finite quantum corrections to U, due to (4 d) 2 (4 d).

8 7 ] The scale-invariant one-loop potential U(φ,σ) = V(φ,σ)+ 1 64π 2 { s=φ,σ Ms(φ,σ) 4 ln M2 s(φ,σ) µ 2 (φ,σ) 3 ] } + U(φ,σ) 2 U = 4 µ 2 { V (µµ φφ µ 2 φ)v φφ +2(µµ φσ µ φ µ σ )V φσ +(µµ σσ µ 2 σ)v σσ ] + 2µ(µ φ V φφ +µ σ V φσ )V φ +2µ(µ φ V φσ +µ σ V σσ )V σ }, µ α = µ α, µ αβ = 2 µ α β, with α,β = φ,σ. If µ = µ(σ) only: U = 4 µ(σ) 2 { 2σ ( V σ V σσ +V φ V φσ ) V Vσσ } If µ=constant, U = 0. On tree-level ground state: U = 0. U: new, one-loop finite correction, beyond the Coleman-Weinberg term.

9 8 ] The scale-invariant one-loop potential. Minimal case: µ = z σ, z: constant. µ = zσ 2/(d 2) ] U = λ φλ m φ 6 σ 2 ( 16λ φ λ m +6λ 2 m 3λ φ λ σ ) φ 4 ( 16λ m +25λ σ ) λm φ 2 σ 2 21λ 2 σσ 4 - U contains higher dimensional operators. It is independent of the subtraction parameter z! - total U is unstable if λ m < 0, due to λ m φ 6 /σ 2 < 0! Higher orders can stabilize it. - if λ 2 m = λ φ λ σ, λ m < 0 for tree-level EWSB, then: U = λ m λ φ ( φ 2 σ 2 + λ m λ φ ) (λ 2 φ φ 4 4λ φ (4λ φ +λ m )φ 2 σ 2 21λ 2 mσ 4) - U can be Taylor expanded: σ = σ +δσ, δσ = quantum fluctuation - spectrum at quantum level: massive φ and a massless dilaton σ (Goldstone) - flat direction - can only predict the ratio φ / σ. Potential unstable under quantum fluctuations. Higher orders may stabilize it. Quantum effective operators present, with known, finite coefficient, independent of z.

10 9 ] Minimizing the one-loop U: λ φ λ m λ σ, and µ = zσ. (*) U = λ φ 4 φ4 + λ m 2 φ2 σ 2 + λ σ 4 σ π 2 { s=1,2 M 4 s ln M2 s z 2 σ 3 ] 2 2 φ 6 + λ φ λ m σ ( ) } 16λ 2 φ λ m +6λ 2 m 3λ φ λ σ φ 4 16λ 2 mφ 2 σ 2 +O(λ 3 m) min: ρ φ 2 σ 2 = λ m λ φ 1 6λ φ 64π 2 ( 4ln3λφ 17/3 )] +O(λ 2 m) m 2 φ = (U φφ +U σσ ) min ; δm 2 φ = 1 64π 2 ( Uφφ + U σσ )min δm 2 φ = σ 2 4λ 2 32π m(4+13ρ)+18λ 2 σ (7λ σ λ φ ρ)+λ m 25λσ (1+ρ) 3λ φ ρ( 32+5ρ+ρ 2 ) ]] λ 2 m σ 2 - fixing the dimensionless subtraction parameter: take z = φ / σ µ = φ, as usual. No tuning needed beyond (*) to keep δm 2 φ σ 2. No dangereous λ φ σ 2. may hold to all orders Callan-Symanzik: z du/dz = 0. see related work of C. Tamarit 2014]

11 10 ] Restrictions on other expressions for µ(σ, φ): Adding a term: L G = 1 2 (ξ φφ 2 +ξ σ σ 2 )R, needed in some models to generate the Planck scale Shaposhnikov et al 2009] µ = z ( ξ φ φ 2 +ξ σ σ 2) 1/2 ] Then: U = (ξ φ φ 2 +ξ σ σ 2 ) (21λ 2 φ ξ φ +λ m ξ σ )ξ φ λ φ φ 8 +(21λ σ ξ σ +λ m ξ φ )ξ σ λ σ σ 8 + negative coefficients of φ 8, σ 8 for λ 2 m = λ φ λ σ. U unstable at large fields. ] U = 3 λm =0 λ2 φ ξ φ φ 6 9ξ σ σ 2 +7 ξ φ φ 2 (ξ φ φ 2 +ξ σ σ 2 ) 2 in the classical decoupling limit: non-decoupling quantum effects, unless σ More general case of: µ(φ,σ) = zσ exp h(φ/σ) ] - similar conclusion. The form of µ(φ,σ) is restricted to avoid such non-decoupling effects Minimal µ = µ(σ) only!

12 11 ] Summary - scale invariance often used to address the hierarchy problem but all regularizations break explicitly the symmetry one wants to study at quantum level. we studied a scale-invariant regularization, with spontaneous breaking of this symmetry. Implications: One-loop scale invariant scalar potential U. U: new correction to U, beyond Coleman-Weinberg term ( evanescent origin). U: independent of subtraction parameter; U φ 6 /σ 2 finite, effective operator(s), destabilize U at large φ. mass correction to φ under control at one-loop (no extra tuning needed). next: study the scalar potential for scale invariant SM (+ dilaton). Non-renormalizability? applications to theories in which preserving scale invariance at loop level needed (CFT s,...)

13 12 ] Scale invariant Standard Model one-loop potential: Ṽ = µ 4 d V, V = λ φ H 4 +λ m H 2 σ 2 + λ σ 4 σ4 ; H = (0,φ)/ 2. M 2 G = λ φ φ 2 +λ m σ 2, M 2 φ,m 2 σ M 2 W = 1 4 g2 φ 2, M 2 Z = 1 4 (g2 +g 2 )φ 2, M 2 t = 1 2 y2 tφ 2. U = λ φ 4 φ4 + λ m 2 φ2 σ 2 + λ σ 4 σ4 + 1 { 3 64π 2 2 (λ φφ 2 +λ m σ 2 ) 2 ln λ σφ 2 +λ m σ 2 3 ] z 2 σ 2 2 φ 6 + λ φ λ m σ ( ) 16λ 2 φ λ m +6λ 2 m 3λ φ λ σ φ 4 ( ) 16λ m +25λ σ λm φ 2 σ 2 21λ 2 σσ 4 + Ms 4 ln M2 s z 2 σ 3 ] (9λ 2 2 φ +λ 2 2 m)φ 4 +2λ m (3λ φ +4λ m +3λ σ )φ 2 σ 2 +(λ 2 m +9λ 2 σ)σ 4 s=φ,σ g4 φ 4 ln g2 2φ 2 4z 2 σ ] (g2 +g 2 ) 2 φ 4 ln g2 φ 2 ] 4z 2 σ 2 5 3φ 4 yt 4 ln φ2 y 2 ]} t 6 2z 2 σ Phenomenology?

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