Toshifumi Noumi (Hong Kong U of Science & Technology, RIKEN - PowerPoint PPT Presentation

@Nambu workshop, November 17th 2015 Spacetime Symmetry and Nambu-Goldstone Bosons mainly based on arXiv:1412.5601 (published in PRD) with Y. Hidaka (RIKEN) and G. Shiu (Wisconsin&HKIAS) Toshifumi Noumi (Hong Kong U of Science & Technology, RIKEN iTHES)

1. Introduction

symmetry breaking in physics

spacetime symmetry breaking condensed matter Poincare symmetry cosmology time-translation

Nambu-Goldstone (NG) modes

Nambu-Goldstone theorem breaking of continuum symmetry gapless mode (NG mode) nonlinear rep of broken sym ＃internal symmetry in relativistic system one broken symmetry one gapless mode with ex. pions as pseudo Nambu-Goldstone bosons cf. recent progress in nonrelativistic case → Watanabe-san’s talk dispersion relation ω = k

there is yet another story for spacetime symmetry breaking

mismatch in NG mode counting ex.1 string [Low-Manohar ’02] broken symmetries: rotation & translation (string position) ex.2 conformal symmetry breaking fig. from Low-Manohar ’02 conformal symmetry Poincare symmetry only one NG mode: dilaton (for dilatation) ※ only one NG mode P µ , L µ ν , D, K µ P µ , L µ ν , D, K µ D : dilatation K µ : special conformal

ex. smectic A phase of liquid crystals - is massless because of original translation symmetry - is massive massive modes for broken symmetries NG mode for translation: position of layers NG mode for rotations: rotations of molecules - translation and rotational symmetries are broken ※ original rotation symmetry transforms also π ξ ˆ i (ˆ i = 1 , 2) π ※ invariance under π → π + const. ξ ˆ i π ξ 1 → ξ 1 + ω 13 , π → π − ω 13 x ( ω 13 : const.)

subtleties even in identification of excitations for broken spacetime symmetries “NG mode” for broken spacetime symmetry massless massive no corresponding dof (unphysical, redundant)

massless massive no corresponding dof (unphysical, redundant) in the low-energy effective theory... “NG mode” for broken spacetime symmetry

in the low-energy effective theory... massless no corresponding dof (unphysical, redundant) “NG mode” for broken spacetime symmetry m . E

in the low-energy effective theory... massless no corresponding dof (unphysical, redundant) “NG mode” for broken spacetime symmetry important to understand what is physical, what is massive, ... m . E

what I discuss today: - global vs local viewpoints of spacetime symmetry - EFT from local symmetry point of view - revisit coset construction based on global symmetry breaking main message: gauging is useful! global symmetry translation, Lorentz, conformal, ... local symmetry diffs, local Lorentz, local Weyl, ... gauge! - identification of physical excitations for broken symmetries

plan of my talk: 1. Introduction 3. EFT from gauging 2. Global vs local 4. Prospects ✔

2. Global symmetry vs local symmetry Q. why # of broken symmetries ≠ # of physical excitations in spacetime symmetry case? global in a nonlinear rep

scalar vs nonzero spin brane scalar brane nonzero spin - same global symmetry breaking pattern: translation in z-direction & rotation in x-z, y-z surfaces - different excitations associated with symmetry breaking in nonlinear rep of broken symmetries y x z

ex. rotation = coord transf + local rotation = = local rotation + acts on all the local field nonzero spin fields local translation around origin global rotation + y y y x x x

rotation symmetry breaking local rotation global rotation around origin local translation = +

rotation symmetry breaking local rotation global rotation around origin local translation = + physical excitations ⇄ broken local symmetries

local decomposition broken by inhomogeneous condensation nonzero spin dimensionful spacetime symmetry local ＝ translation Lorentz rescaling local local in relativistic system ＝ translation ＋ Lorentz ＋ re

local decomposition ＝ translation Lorentz rescaling dilatation special conformal ex. conformal symmetry → Poincare symmetry local broken by local local spacetime symmetry dimensionful nonzero spin condensation inhomogeneous in relativistic system ＝ translation ＋ Lorentz ＋ re ✔ ✔ ✔ ✔ ✔

local decomposition ＝ translation Lorentz rescaling dilatation special conformal ex. conformal symmetry → Poincare symmetry local broken by local local spacetime symmetry dimensionful nonzero spin condensation inhomogeneous ※ only one broken symmetry in local viewpoint ⇄ dilaton in relativistic system ＝ translation ＋ Lorentz ＋ re ✔ ✔ ✔ ✔ ✔

local decomposition local to identify excitations for broken symmetries local symmetry viewpoint is useful ※ only one broken symmetry in local viewpoint ⇄ dilaton dilatation special conformal broken by local ex. conformal symmetry → Poincare symmetry ＝ translation Lorentz rescaling local spacetime symmetry dimensionful nonzero spin condensation inhomogeneous → local decomposition by gauging spacetime symmetry! in relativistic system ＝ translation ＋ Lorentz ＋ re ✔ ✔ ✔ ✔ ✔

gauging spacetime symmetry spacetime symmetry local ＝ translation Lorentz rescaling local local in relativistic system ＝ translation ＋ Lorentz ＋ re

gauging spacetime symmetry gauged 1. Weyl gauging (always possible): introduce a local Weyl invariant curved spacetime action 2. Ricci gauging (only when conformal): diffeomorphism ＋ local Lorentz ＋ local Weyl in relativistic system spacetime symmetry ※ vierbein aa & local Lorentz frame metric aa & curved coord by introducing ∈ e m g µ ν µ introducing a Weyl gauge field W µ

Q. why # of broken symmetries ≠ # of physical excitations in spacetime symmetry case? A. spacetime symmetry coordinate transformation → local decomposition by gauging spacetime symmetry physical excitations ⇄ broken local symmetries 3

plan of my talk: 1. Introduction 3. EFT from gauging 2. Global vs local 4. Prospects ✔ ✔

3. EFT from gauging - internal symmetry case - EFT for diffs breaking - massive modes for broken symmetries - relation to coset construction

EFT from massive gauge bosons

unitary gauge action - NG modes are eaten by gauge boson # effective action for massive gauge boson : - : gauge coupling, : order parameter, mass = with - residual H gauge symmetry # unitary gauge is convenient to construct effective action - global symmetry limit = massless limit is singular broken symmetry residual symmetry # consider an internal symmetry breaking G → H ( h : g = h ⊕ a a : 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 gv g v

Stuckelberg method ※ is the broken symmetry part of # in the global symmetry limit , gauge sector NG mode sector - the gauge sector decouples from NG mode π (broken symmetry) with # introduce NG modes by field dependent gauge transf µ = Ω � 1 A µ Ω + Ω � 1 ∂ µ Ω A µ → A 0 Ω = e π a ( x ) T a T a ∈ a - NG modes = coordinates of G/H π a gauge coupling g J µ = Ω − 1 ∂ µ Ω A µ g → 0 − v 2  � Z 2 J a µ J a µ + . . . d 4 x tr S = J µ = Ω − 1 ∂ µ Ω J a µ ∈ a

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

unitary gauge for broken diffs V ( φ ) φ φ + z 0 φ φ − φ + φ − h φ ( x ) i = ¯ φ ( z )

- translation symmetry → is also stable unitary gauge for broken diffs V ( φ ) φ φ + z 0 − c φ φ − φ + φ − h φ ( x ) i = ¯ φ ( z ) h φ ( x ) i = ¯ φ ( z + c )

unitary gauge for broken diffs - translation symmetry → ※ can be absorbed by coordinate shift V ( φ ) φ φ + z 0 z 0 − c φ φ − φ + φ − h φ ( x ) i = ¯ φ ( z ) h φ ( x ) i = ¯ φ ( z + c ) = ¯ φ ( z 0 ) z 0 = z + c

- translation symmetry → ※ can be absorbed by coordinate shift unitary gauge for broken diffs V ( φ ) φ φ + z 0 z 0 − c φ φ − φ + φ − h φ ( x ) i = ¯ φ ( z ) h φ ( x ) i = ¯ φ ( z + c ) = ¯ φ ( z 0 ) z 0 = z + c - NG mode π: h φ ( x ) i = ¯ � � z + π ( x ) = φ

※ can be absorbed by coordinate shift - translation symmetry → unitary gauge for broken diffs V ( φ ) φ φ + z 0 z 0 − c φ φ − φ + φ − h φ ( x ) i = ¯ φ ( z ) h φ ( x ) i = ¯ φ ( z + c ) = ¯ φ ( z 0 ) z 0 = z + c - NG mode π: h φ ( x ) i = ¯ = ¯ � � φ ( z 0 ) z + π ( x ) φ ※ can be absorbed by coordinate transf z 0 = z + π ( x )