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

• ne broken symmetry
• ne gapless mode with

dispersion relation ω = k

• ex. pions as pseudo Nambu-Goldstone bosons
• cf. recent progress in nonrelativistic case → Watanabe-san’s talk

there is yet another story for spacetime symmetry breaking

mismatch in NG mode counting

ex.1 string [Low-Manohar ’02]

broken symmetries: rotation & translation

※ only one NG mode

(string position)

ex.2 conformal symmetry breaking

• fig. from Low-Manohar ’02

Pµ, Lµν, D, Kµ D : dilatation Kµ : special conformal

conformal symmetry

Pµ, Lµν, D, Kµ

Poincare symmetry

• nly one NG mode:

dilaton (for dilatation)

• ex. smectic A phase of liquid crystals
• translation and rotational symmetries are broken

π

• is massive

massive modes for broken symmetries

NG mode for translation: position of layers NG mode for rotations: rotations of molecules

ξˆ

i

(ˆ i = 1, 2)

• is massless because of original translation symmetry

π

※ invariance under π → π + const.

ξˆ

i

※ original rotation symmetry transforms also

π

ξ1 → ξ1 + ω13, π → π − ω13 x (ω13 : 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)

m . E

“NG mode” for broken spacetime symmetry

in the low-energy effective theory...

massless no corresponding dof

(unphysical, redundant)

m . E

“NG mode” for broken spacetime symmetry

important to understand what is physical, what is massive, ...

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

x y z

• ex. rotation = coord transf + local rotation

local rotation global rotation around origin local translation nonzero spin fields all the local field = + acts on

x x

y

x

y y

= +

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 in relativistic system ＝ translation ＋ Lorentz ＋ re

local ＝ translation Lorentz rescaling local local

local decomposition

broken by inhomogeneous condensation nonzero spin dimensionful

spacetime symmetry in relativistic system ＝ translation ＋ Lorentz ＋ re

local ＝ translation Lorentz rescaling local local

• ex. conformal symmetry → Poincare symmetry

special conformal dilatation ✔ ✔ ✔ ✔ ✔

local decomposition

broken by inhomogeneous condensation nonzero spin dimensionful

spacetime symmetry in relativistic system ＝ translation ＋ Lorentz ＋ re

local ＝ translation Lorentz rescaling local local

• ex. conformal symmetry → Poincare symmetry

special conformal dilatation ✔ ✔ ✔ ✔ ✔

※ only one broken symmetry in local viewpoint ⇄ dilaton

local decomposition

broken by inhomogeneous condensation nonzero spin dimensionful

spacetime symmetry in relativistic system ＝ translation ＋ Lorentz ＋ re

local ＝ translation Lorentz rescaling local local

• ex. conformal symmetry → Poincare symmetry

special conformal dilatation ✔ ✔ ✔ ✔ ✔

※ only one broken symmetry in local viewpoint ⇄ dilaton

local symmetry viewpoint is useful to identify excitations for broken symmetries → local decomposition by gauging spacetime symmetry!

gauging spacetime symmetry

spacetime symmetry in relativistic system ＝ translation ＋ Lorentz ＋ re

local ＝ translation Lorentz rescaling local local

gauging spacetime symmetry

gauged by introducing curved coord & metric aa

gµν

local Lorentz frame & vierbein aa

em

µ

spacetime symmetry in relativistic system

diffeomorphism ＋ local Lorentz ＋ local Weyl

∈

• 2. Ricci gauging (only when conformal):

introduce a local Weyl invariant curved spacetime action

• 1. Weyl gauging (always possible):

introducing a Weyl gauge field Wµ

※

• Q. why

# of broken symmetries ≠ # of physical excitations in spacetime symmetry case?

3

• A. spacetime symmetry coordinate transformation

→ local decomposition by gauging spacetime symmetry physical excitations ⇄ broken local symmetries

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

# consider an internal symmetry breaking G → H

g = h ⊕ a ( h: a :

residual symmetry broken symmetry

• NG modes are eaten by gauge boson

# unitary gauge is convenient to construct effective action

• residual H gauge symmetry

with

Z d4x tr  − 1 4g2 F µνFµν − v2 2 Aa µ Aµ

a + . . .

• Aaµ ∈ a
• : gauge coupling, : order parameter, mass =

g v

gv

Aµ

# effective action for massive gauge boson :

• global symmetry limit = massless limit is singular

Stuckelberg method

Aµ → A0

µ = Ω1AµΩ + Ω1∂µΩ

# introduce NG modes by field dependent gauge transf

πa

• NG modes = coordinates of G/H

with (broken symmetry)

Ω = eπa(x)Ta Ta ∈ a

• NG mode sector

Jµ = Ω−1∂µΩ

gauge sector

Aµ

gauge coupling g

# in the global symmetry limit , the gauge sector decouples from NG mode π

g → 0

Jµ = Ω−1∂µΩ

※ is the broken symmetry part of

Jaµ ∈ a

S = Z d4x tr  −v2 2 JaµJaµ + . . .

• 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

φ z

φ− φ+

φ

φ+ φ− V (φ)

hφ(x)i = ¯ φ(z)

unitary gauge for broken diffs

φ z

φ− φ+

φ

φ+ φ− V (φ)

hφ(x)i = ¯ φ(z)

• translation symmetry → is also stable

hφ(x)i = ¯ φ(z + c)

unitary gauge for broken diffs

−c

φ z

φ− φ+

φ

φ+ φ− V (φ)

hφ(x)i = ¯ φ(z)

unitary gauge for broken diffs

−c

z0 = z + c

※ can be absorbed by coordinate shift

z0

• translation symmetry →

hφ(x)i = ¯ φ(z + c) = ¯ φ(z0)

φ z

φ− φ+

φ

φ+ φ− V (φ)

hφ(x)i = ¯ φ(z)

unitary gauge for broken diffs

−c

z0 = z + c

※ can be absorbed by coordinate shift

• NG mode π:hφ(x)i = ¯

φ

• z + π(x)
• =

z0

• translation symmetry →

hφ(x)i = ¯ φ(z + c) = ¯ φ(z0)

φ z

φ− φ+

φ

φ+ φ− V (φ)

hφ(x)i = ¯ φ(z)

unitary gauge for broken diffs

−c

z0 = z + c

※ can be absorbed by coordinate shift

• NG mode π:hφ(x)i = ¯

φ

• z + π(x)
• = ¯

φ(z0)

※ can be absorbed by coordinate transf z0 = z + π(x)

z0

• translation symmetry →

hφ(x)i = ¯ φ(z + c) = ¯ φ(z0)

φ z

φ− φ+

φ

φ+ φ− V (φ)

hφ(x)i = ¯ φ(z)

unitary gauge for broken diffs

−c

z0 = z + c

※ can be absorbed by coordinate shift

• NG mode π:hφ(x)i = ¯

φ

• z + π(x)
• = ¯

φ(z0)

※ can be absorbed by coordinate transf z0 = z + π(x)

z0

• translation symmetry →

hφ(x)i = ¯ φ(z + c) = ¯ φ(z0)

※ gauging = introduction of general coordinate system ※ unitary gauge condition: φ(z) = ¯

φ(z)

※ residual 2+1 dim diffeomorphism symmetry

unitary gauge action & Stuckelberg method

dof = metric , residual symmetry = (2+1)-dim diffeo

gµν

• unitary gauge action (cf. EFT for inflation [’07 Cheung et al.])

S = Z d4x√−g ⇥ α(z) + β(z)gzz(x) + γ(z)(gzz − 1)2 + . . . ⇤

※ α, β, γ depend on details of microscopic theory

• action for NG modes
• 1. Stuckelberg method: z → z + π(x)
• 2. decouple the gauge sector ⇔ to set gµν = ηµν

※ background (bulk) eom is used

α(z) = β(z) S = Z d4x ⇥ α(z)∂µπ∂µπ + 4γ(z)(∂zπ)2 + O(π3) ⇤ Z ⇥ ⇤

α(z) → α(z + π), gzz → gzz + 2∂zπ + (∂µπ)2

e.g.

• QCD 相図

a

• 20
• 10

• 1.0
• 0.5

z

¯ ψψ(z)

＃ second order action for diffs breaking

let us take a closer look at the simplest action

S2 = Z d4x α(z)∂µπ∂µπ

• free function : nonzero where translation is broken

α(z)

z

α(z)

L

• utside the brane:

no kinetic term → no NG mode

※ Nambu-Goto action in low energy limit

• ex. 1 domain wall
• ex. 2 periodic modulations [cf. ’15 Hidaka-Kamikado-Kanazawa-TN]

→ instability @ finite temp in large volume

E ∼ 0 · k2

x,y + Bk4 x,y + Ck2 z

a nonrelativistic analogue shows dispersion is strongly anisotropic

massive modes associated with broken symmetries

＃ nonzero spin brane

z t, x, y

in global sense: in local sense: translation and Lorentz invariance are broken z-diffeo & local Lorentz are broken

z-µ

※ introduce and to gauge spacetime symmetry

gµν em

µ

→ translation modes and spin modes can be eaten dynamical dof: metric , vierbein

gµν em

µ

residual symmetry: (1+2)-dim diffeo x local Lorentz ※ in such a unitary gauge, symmetry breaking

• : breaks the time diffs

S = SP + SL + SP L

decompose action schematically as

＃ effective action

• : breaks the local Lorentz

SL

SL SP = Z d4x ⇥ α(∂µπ)2 + γ(∂zπ)2 + O(π3) ⇤

SP

S = 1 2 Z d4x h α1

• rµe3

ν

2 + α2

• rµe3

ν

rνeµ3 + α3

• eµ

3rµe3 ν

2i

※ kinetic terms forπ ※ invariant under the shift π → π + const.

• : breaks the time diffs

S = SP + SL + SP L

decompose action schematically as

＃ effective action

• : breaks the local Lorentz

SL

SL SP = Z d4x ⇥ α(∂µπ)2 + γ(∂zπ)2 + O(π3) ⇤

SP

= −1 2 Z d4x h α1(∂b

µξb ν)2 + α2(∂b µξb ν)(∂b νξb µ) + (α1 + α3)(∂zξb ν)2i

※ kinetic terms forπ ※ invariant under the shift π → π + const.

• : breaks the time diffs

S = SP + SL + SP L

decompose action schematically as

＃ effective action

• : breaks the local Lorentz

SL

SL SP = Z d4x ⇥ α(∂µπ)2 + γ(∂zπ)2 + O(π3) ⇤

SP

= −1 2 Z d4x h α1(∂b

µξb ν)2 + α2(∂b µξb ν)(∂b νξb µ) + (α1 + α3)(∂zξb ν)2i

※ kinetic terms forπ ※ invariant under the shift π → π + const. ※ kinetic terms for spin modes ξb

µ (b

µ = t, x, y)

※ invariant under the shift ξb

µ → ξb µ + const.

S = SP + SL + SP L

decompose action schematically as

＃ effective action

SP L

• : breaks both of diffs & local Lorentz

SP L = Z d4x√−gm2(eµ

3nµ − 1) →

Z d4x  −m2 2 (ξb

µ − ∂b µπ)2 + . . .

• ※ mass term for and mixing of &

ξb

µ

ξb

µ

π

※ invariant under the global rotation

ξ1 → ξ1 + ω13, π → π − ω13 x (ω13 : const.)

mass term is not prohibited by symmetry essentially because global rotation transforms also

π

more generally, NG fields for local Lorentz and local Weyl become massive when they are broken at the same time as diffeomorphism

massless massive

P

L, D

EFT from gauging

• 1. gauging spacetime symmetries
• 2. take the unitary gauge (broken symmetry modes are eaten)
• 3. unitary gauge action based on residual symmetry
• 4. Stuckelberg method & decouple the gauge sector

in contrast to the internal symmetry case,

• coefficient functions of coord when diffs are broken
• massive mode associated with broken symmetries
• local picture is important to identify such massive modes

relation to coset construction

• introduced fields for all the broken global symmetries

πa

• have a mass term when contains broken generators

πa [Pµ, Ta] (f b

aPµ 6= 0)

• if we are interested in the massless sector only,

we may impose the so-called inverse Higgs constraints generically include massive & unphysical ones

f b

aPµ 6= 0

for to remove massive and unphysical ones

ab

µ ∼ ∂µπb + f b aPµπa + . . . = 0

EFT from gauging

physical ones only

coset construction

include unphysical ones to realize broken symmetry inverse Higgs constraints integrate out massive & unphysical ones

EFT from gauging

physical ones only

coset construction

include unphysical ones to realize broken symmetry inverse Higgs constraints integrate out massive & unphysical ones

EFT based on local symmetry viewpoint is

• more economical in particular when including massive modes
• convenient to apply gravitational systems such as cosmology

• 4. Prospects

# massive NG modes during inflation? [to appear w/Delacretaz-Senatore]

• cosmological collider physics (named by Arkani Hamed-Maldacena)

※ massive fields with mass affects inflation dynamics

m . H

• one realization via time diffs + local boost breaking

mass can be read off from primordial non-Gaussianity

[Chen-Wang, Baumann-Green, TN-Yamaguchi-Yokoyama, Arkani Hamed-Maldacena]

# test of isotropy during inflation [in progress w/Gong-Shiu-Soda-Yamaguchi]

• how accurate our assumption of isotropic universe?

※ effective action is constructed similarly to nonzero spin brane case

˙ σ

※ anisotropy is characterized by ~ rotation symmetry breaking scale

ds2 = −dt2 + a2e2σ(dx2 + dy2) + a2e−4σdz2

anirotropic background:

※ anisotropy of primordial spectrum, primordial gravitational waves, ...

prospects1: applications to cosmology

・hydrodynamics from spacetime symmetry viewpoint

• finite temperatures, finite densities, ...

・more on nonrelativistic case ・Wess-Zumino term vs gauge anomaly

• ther directions:

Thanks!

・global vs local picture of spacetime symmetry breaking

• # of physical excitations ≠ # of broken global symmetries
• physical excitations ⇄ broken local symmetries

・EFT by gauging spacetime symmetries

• correct identification of physical excitations
• spacetime symmetry diffeo x local Lorentz x (an)isotropic Weyl

∈

• unitary gauge & Stuckelberg method
• classification of physical meaning of inverse Higgs constraints

・coset construction revisited

① remove redundant NG fields ② integrate out massive mode associated with broken symmetry

gauging is useful for spacetime symmetry breaking!