＠YITP workshop July 22nd 2014 Toshifumi Noumi (Math Phys Lab, RIKEN) Effective Field Theory for Spacetime Symmetry Breaking based on a paper in preparation with Y. Hidaka (RIKEN) and G. Shiu (HKUST)
1. Introduction
symmetry breaking in physics
spacetime symmetry breaking condensed matter cosmology
# various phases of liquid crystal figs: wikipedia nematic fig: Basar Dunne ’08 chiral spiral (complex) real kink # inhomogeneous chiral condensate (in QCD phase diagram?) smectic C smectic A h ¯ ψψ i + i h ¯ ψ i γ 5 ψ i = ∆ ( z ) ∆ 1.0 0.5 z - 20 - 10 10 20 - 0.5 - 1.0
# various phases of liquid crystal chiral spiral (complex) - # of gapless modes? their dispersion relations? would like to understand low-energy dynamics nematic real kink fig: Basar Dunne ’08 - from (spacetime) symmetry point of view? figs: wikipedia # inhomogeneous chiral condensate (in QCD phase diagram?) smectic A smectic C h ¯ ψψ i + i h ¯ ψ i γ 5 ψ i = ∆ ( z ) ∆ 1.0 0.5 z - 20 - 10 10 20 - 0.5 - 1.0
# cosmology - cosmic expansion breaks time translation generically - various models for inflation ex. anisotropic inflation: rotation is also broken ex. gaugeflation: internal SU(2) x rotation → diagonal SU(2)
# cosmology - cosmic expansion breaks time translation generically - various models for inflation ex. anisotropic inflation: rotation is also broken ex. gaugeflation: internal SU(2) x rotation → diagonal SU(2) cosmological models from symmetry viewpoint - curved background, gravitational theory - massive fields with are also relevant m . H
coset construction
＃ coset construction for internal symmetry breaking - ingredients of effective action: ※ coset construction provides general effective action broken symmetry residual symmetry - effective action is local right H invariant (broken symmetry) with consider an internal symmetry breaking G → H ( h : g = h ⊕ a a : - NG modes = coordinates of G/H π a Ω = e π a ( x ) T a T a ∈ a Maurer-Cartan one form J µ = Ω − 1 ∂ µ Ω
＃ extension to spacetime symmetry breaking - global symmetry picture leads to wrong NG mode counting broken symmetry: dilatation and special conformal with - remove by imposing the inverse Higgs constraints ex. conformal symmetry breaking (conformal → Poincare) ※ NG modes = local transformations of order parameters : spurious field to be removed : dilaton, - introduce two types of “NG modes” K µ D MC form: J µ = Ω − 1 ∂ µ Ω Ω = e x µ P µ e φ D e ξ µ K µ φ ξ µ ξ µ
motivation coset construction: - has been applied to various condensed matter systems - captures a certain aspects of spacetime symmetry breaking however, its understanding seems not complete - no proof that coset construction provides general action - appearance of spurious NG mode may not be attractive would like to have an approach - without spurious NG mode from the beginning - appropriate to curved spacetime & gravitational theory → effective theory based on a local symmetry picture
plan of my talk: 1. Introduction 3. Case study 1: scalar domain walls 2. Basic strategy 4. Case study 2: vector domain walls 5. Summary and discussion ✔
2. Basic strategy
coset construction from gauge symmetry breaking
effective action for massive gauge boson : - : gauge coupling, : order parameter introduce NG modes by Stuckelberg method: dynamical dof = gauge field only - NG modes are eaten by gauge boson (unitary gauge) with with A µ 4 g 2 F µ ν F µ ν − v 2 � Z − 1 d 4 x tr 2 A a µ A µ a + . . . A a µ ∈ a g v Ω = e π a ( x ) T a µ = Ω � 1 A µ Ω + Ω � 1 ∂ µ Ω A µ → A 0 ※ global symmetry limit can be obtained by setting A µ = 0 A µ → J µ = Ω − 1 ∂ µ Ω in the unitary gauge effective action 4 g 2 F µ ν F µ ν − v 2 − v 2 � � Z Z − 1 d 4 x tr d 4 x tr 2 A a µ A µ 2 J a µ J µ a + . . . a + . . . →
gauge field is the only dof in the unitary gauge can be thought of as a proof for completeness of coset construction unitary gauge is convenient to find general ingredients for EFT the most careful way to construct the general effective action will be 1. gauge the (broken) global symmetry 2. write down the unitary gauge effective action 3. introduce NG modes by Stuckelberg method and decouple the gauge sector
local properties of spacetime symmetry
＃ local properties of spacetime symmetry - 2nd term: deformations of coord. system locally, a combination of Poincare & isotropic rescaling ex. special conformal on Minkowski space ・symmetric traceless : anisotropic rescaling ・antisymmetric : Lorentz transformation ・trace part : isotropic rescaling - 1st term: shift of coord. system (translation) its local properties around a point can be read off as consider a spacetime symmetry associated with x 0 µ = x µ − ✏ µ ( x ) x µ = x µ ∗ ✏ µ ( x ) = ✏ µ ( x ∗ ) + ( x ν � x ν ∗ ) r ν ✏ µ ( x ) + . . . r µ ✏ ν = � ν ν + ! µ µ � + s µ ν λ s µ ν ω µ ν r µ ✏ ν = 2 � ν µ ( b · x ) + 2( b µ x ν � b ν x µ )
in diffeomorphism, local Lorentz, local (an)isotropic Weyl as the local decomposition suggests, any spacetime symmetry transformation can be embedded relativistic symmetry di ff eomorphism local Lorentz local Weyl translation � isometry � � conformal � � � Table 1: Embedding of spacetime symmetry in relativistic systems. nonrelativistic symmetry foliation preserving local rotation local anisotropic Weyl translation � Galilean � � Schrodinger � � � Galilean conformal � � � Table 2: Embedding of spacetime symmetry in nonrelativistic systems.
gauging spacetime symmetry
＃ gauging spacetime symmetry - diffeo & local Lorentz can be gauged by introducing curved spacetime action global spacetime symmetry diffeo x local Lorentz x local Weyl - Weyl symmetry 1. Ricci gauging (not necessarily possible) introduce a local Weyl invariant curved spacetime action 2. Weyl gauging (always possible) ∈ d 4 x p� g L [ Φ , e µ Z Z d 4 x L [ Φ , ∂ m Φ ] ! m r µ Φ ] gauge global Weyl symmetry by introducing a gauge field W µ
EFT recipe
we construct the effective action in the following way: symmetry breaking pattern based on local symmetries: once symmetry breaking patterns are given or identified, and decouple the gauge sector 3. introduce NG modes by Stuckelberg method 2. write down the unitary gauge effective action 1. gauge the (broken) global symmetry di ff eomorphism local Lorentz local Weyl internal gauge spacetime dependence spin scaling dimension internal charge vierbein e m metric g µ ν Weyl gauge field W µ gauge field A µ µ = ¯ Φ A ( x ) Φ A ( x ) can be classified by condensation patterns ⌦ ↵
plan of my talk: 1. Introduction 3. Case study 1: scalar domain walls 2. Basic strategy 4. Case study 2: vector domain walls 5. Summary and discussion ✔ ✔
3. Scalar domain-walls
＃ domain-wall configurations of a real scalar in global sense: single domain wall translation and Lorentz invariance are broken symmetry breaking multi domain walls φ φ 1.0 1.0 0.5 0.5 z z - 20 - 10 10 20 - 20 - 10 10 20 - 0.5 - 0.5 - 1.0 - 1.0
＃ domain-wall configurations of a real scalar in global sense: single domain wall full diffeo → (1+2)-dim diffeo translation and Lorentz invariance are broken in local sense: only z-diffeo is broken symmetry breaking multi domain walls φ φ 1.0 1.0 0.5 0.5 z z - 20 - 10 10 20 - 20 - 10 10 20 - 0.5 - 0.5 - 1.0 - 1.0
＃ domain-wall configurations of a real scalar - unitary gauge action (cf. EFT for inflation [’07 Cheung et al.] ) - action for NG modes dof = metric , residual symmetry = (1+2)-dim diffeo g µ ν Z α ( z ) + β ( z ) g zz ( x ) + γ ( z )( g zz − 1) 2 + . . . d 4 x √− g ⇥ ⇤ S = 1. Stuckelberg method: z → z + π ( x ) 2. decouple the gauge sector ⇔ to set g µ ν = η µ ν Z d 4 x [ α ( z + π ) + β ( z + π )(1 + 2 ∂ z π + ∂ µ π∂ µ π ) S = + γ ( z + π )(2 ∂ z π + . . . ) 2 + . . . ⇤ 3. background (bulk) eom → α ( z ) = β ( z )
＃ domain-wall configurations of a real scalar dof = metric , residual symmetry = (1+2)-dim diffeo - unitary gauge action (cf. EFT for inflation [’07 Cheung et al.] ) - action for NG modes g µ ν Z α ( z ) + β ( z ) g zz ( x ) + γ ( z )( g zz − 1) 2 + . . . d 4 x √− g ⇥ ⇤ S = 1. Stuckelberg method: z → z + π ( x ) 2. decouple the gauge sector ⇔ to set g µ ν = η µ ν Z α ( z ) ∂ µ π∂ µ π + 4 γ ( z )( ∂ z π ) 2 + O ( π 3 ) d 4 x ⇥ ⇤ S = Z ⇤ z = ∞ d 3 x α ( z ) π + O ( π 2 ) ⇥ + z = −∞ 3. background (bulk) eom → α ( z ) = β ( z )
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