Aspects of Spontaneous Lorentz Violation

Robert Bluhm Colby College

IUCSS School on CPT & Lorentz Violating SME, Indiana University, June 2012

Outline:

I. Review & Motivations

II.  Spontaneous Lorentz Violation

III.  Nambu-Goldstone Modes & Higgs Mech.

IV. Examples: Bumblebee & Tensor Models

V. Conclusions

I. Review & Motivations

Previous talk looked at how to construct the SME in the presence of gravity

Lorentz symmetry comes in two varieties:

⇒ 

⇒ 

global symmetry of special relativity

- field theories invariant under global LTs

local symmetry of general relativity

-  Lorentz symmetry holds locally

SME lagrangian   observer scalar formed from tensors,   covariant derivatives, spinors, gamma   matrices, etc. & SME coeffs.

Have 2 symmetries in gravity: •  local Lorentz symmetry •  spacetime diffeomorphisms

SME with Gravity

GR involves tensors on a curved spacetime manifold Tλµν. . . ⇒  spacetime tensor components

Tabc. . . ⇒  local Lorentz frame components

includes gravity, SM, and LV sectors

To reveal the local Lorentz symmetry, introduce local tensor components in Lorentz frames

These components are connected by a vierbein

vierbein: ⇒  relates local and manifold frames

⇒  tetrad of spacetime coord. vectors ⇒  can accommodate spinors

In a vierbein formalism, must also introduce a spin connection

spin connection: appears in cov. derivs. of local tensors ⇒

In Riemann spacetime with (metric)

⇒  spin connection is determined by the vierbein ⇒  not independent degrees of freedom

⇒  no evidence for (or against) torsion ⇒  but should exist if gravity is like a gauge theory

Riemann-Cartan spacetime ⇒ 

Tλµν = Γλµν - Γλνµ

⇒  spin connection becomes dynamically independent ⇒  gives gravity the form of a gauge theory

Can also introduce torsion

New geometry emerges:

curvature = Rκλµν torsion = Tλµν

→ 16 components

→ 24 components

The SME with gravity includes curvature & torsion

Constructing the SME with Gravity

Example: fermion coupled to gravity:

where

Additional fermion couplings might include:

Riemannian limit (zero torsion):

Terms in the pure-gravity sector might include:

For exploring phenomenology, it is useful to start with a minimal model that extends GR (without torsion)

Jay Tasson’s talk will look at phenomenology

explicit breaking incompatible with geometrical identities, but spontaneous symmetry breaking evades this difficulty

Explicit vs. Spontaneous Lorentz Violation (SLV)

SME coeffs. can result from either spontaneous or explicit Lorentz violation

With explicit LV

⇒  act as fixed background fields in any observer frame

No-go theorem:

But with spontaneous LV

⇒  arise as vev’s ⇒  must be treated dynamically

Spontaneous Lorentz Violation (SLV)

Question: What happens if Lorentz symmetry is spontaneously broken in a theory of gravity?

originally motivated from quantum gravity & string theory

General Relativity is a classical theory not compatible with quantum physics

Open Problem

Expect particle physics and classical gravity to merge in a quantum theory of gravity

Planck scale:

Is Lorentz symmetry exact at the Planck scale?

•  Nonpertubative vacuum in string field theory •  Produces vevs for tensor fields

Mechanisms exist in SFT that could lead to vector/tensor fields acquiring nonzero vacuum expectation values (vevs)

⇒  can lead to spontaneous Lorentz violation

•  fundamental theory fully Lorentz invariant •  vacuum breaks Lorentz symmetry •  evades the no-go theorem

⇒  provides most elegant form of Lorentz violation

<Τ> ≠ 0

String Theory & SLV

SME coeffs., e.g., aµ, bµ, cµν, dµν, Hµν, . . . arise as vacuum expectation values when SLV occurs

A symmetry is spontaneously broken when the eqs. of motion obey the symmetry but the solutions do not.

e.g., magnet dipole-dipole ints. are spatially symmetric but when a magnet forms the dipoles align along a particular direction

The rotational symmetry is spontaneously broken

e.g., push on a stick it’s rotationally symmetric but it buckles in a spontaneously chosen direction in space

With SSB, the symmetry is still there dynamically, but is hidden by the solution

II. Spontaneous Lorentz Violation

Spontaneous symmetry breaking occurs in gauge theories

a potential V has a nonzero minimum

e.g., in the electroweak theory, a scalar field has a vacuum solution (vev) that breaks the gauge symmetry

The theory has multiple potential vacuum solutions

the physical vacuum picks one, breaking the symmetry

V

f

V

f

const. scalar field (electroweak)

<f> ≠ 0

<T> ≠ 0 tensor vev

vacuum breaks Lorentz symmetry

In the electroweak theory, the vev is a constant scalar has no preferred directions or rest frame preserves Lorentz symmetry

But what if a vector or tensor field acquires a nonzero vev? there would be preferred directions in spacetime spontaneous breaking of Lorentz symmetry

Consider a Lorentz-invariant lagrangian

Lkinetic =1

16⇥GR� 1

4Bµ�B

µ�

Bµ� = DµB� �D�Bµ

Will-Nordvedt

Lkinetic =1

16⇥G

�

a1R + a2BµBµR + a3BµB�Rµ� + a4DµB�D

µB�

+a5DµB�D�Bµ + a6DµBµD�B

� ⇥

L =1

16⇥G

�

a1R + a2BµBµR + a3BµB�Rµ� + a4DµB�D

µB�

+a5DµB�D�Bµ + a6DµBµD�B

� ⇥

� V (BµB� ± b2) + LM

How is SLV introduced?

include a potential that has a nontrivial minimum that occurs when has a nonzero vev

with tensor fields

Bµ = bµ + Aµ

⌃T ⌥ ⇧= 0

L ⇥ ⌃Mk ⌃T ⌥� ⌦̄(i↵)k

[xµ, x⌥] = i⇧µ⌥

L ⇥ 1

4iq ⇧�⇥ F�⇥ ⌦̄ ⇤

µ Dµ⌦

L⇤L

L = Lgravity + LSM + LLV + · · ·

Ta ⇤ ⇥ ba Tb ⌅ Ta + ⌅ b

a Tb

Tµ ⇤ Tµ � (↵µ��)T� � ��(↵�Tµ)

Bµ = bµ + Aµ

⌃T ⌥ ⇧= 0

L ⇥ ⌃Mk ⌃T ⌥� ⌦̄(i↵)k

[xµ, x⌥] = i⇧µ⌥

L ⇥ 1

4iq ⇧�⇥ F�⇥ ⌦̄ ⇤

µ Dµ⌦

L⇤L

L = Lgravity + LSM + LLV + · · ·

Ta ⇤ ⇥ ba Tb ⌅ Ta + ⌅ b

a Tb

Tµ ⇤ Tµ � (↵µ��)T� � ��(↵�Tµ)

e.g., in flat spacetime, with components

Bµ = bµ + Aµ

⌃T ⌥ ⇧= 0

L ⇥ ⌃Mk ⌃T ⌥� ⌦̄(i↵)k

[xµ, x⌥] = i⇧µ⌥

L ⇥ 1

4iq ⇧�⇥ F�⇥ ⌦̄ ⇤

µ Dµ⌦

L⇤L

L = Lgravity + LSM + LLV + · · ·

Ta ⇤ ⇥ ba Tb ⌅ Ta + ⌅ b

a Tb

Tµ ⇤ Tµ � (↵µ��)T� � ��(↵�Tµ)

eµ⌥ = ⇧µ⌥ + (12hµ⌥ + µ⌥)

hµ⌥ = h⌥µ

µ⌥ = � ⌥µ

⇤Tµ⌥···⌅ ⇥ tµ⌥···

⌅T⌃µ⌥··· = (T⌃µ⌥··· � t⌃µ⌥···)

�⌃µ⌥··· = (T⌃µ⌥··· � t⌃µ⌥···)

T⌃µ⌥···g⌃�gµ⇥g⌥⇤ . . . T�⇥⇤··· = t2

t2 = t⌃µ⌥···⇧⌃�⇧µ⇥⇧⌥⇤ . . . t�⇥⇤···

V = V (T⌃µ⌥···g⌃�gµ⇥g⌥⇤ . . . T�⇥⇤··· � t2)

T⌃µ⌥··· = e �⌃ e ⇥

µ e ⇤⌥ . . . t�⇥⇤···

has a minimum when

eµ⌥ = ⇧µ⌥ + (12hµ⌥ + µ⌥)

hµ⌥ = h⌥µ

µ⌥ = � ⌥µ

⇤Tµ⌥···⌅ ⇥ tµ⌥···

⌅T⌃µ⌥··· = (T⌃µ⌥··· � t⌃µ⌥···)

�⌃µ⌥··· = (T⌃µ⌥··· � t⌃µ⌥···)

T⌃µ⌥···g⌃�gµ⇥g⌥⇤ . . . T�⇥⇤··· = t2

t2 = t⌃µ⌥···⇧⌃�⇧µ⇥⇧⌥⇤ . . . t�⇥⇤···

t2 = tabc···⇧ad⇧be⇧cf . . . tdef ···

V = V (T⌃µ⌥···g⌃�gµ⇥g⌥⇤ . . . T�⇥⇤··· � t2)

where

What about in curved spacetime?

Lorentz symmetry is a local symmetry

Tabc···⇥ � da � e

b � fc Tdef ···

⇤ Tabc··· + ⇤ da Tdbc··· + ⇤ e

b Taec··· + ⇤ fc Tabf ···

T⌅µ⇧ ⇥ T⌅µ⇧�(�⌅⌃�)T�µ⇧�(�µ⌃�)T⌅�⇧�(�⇧⌃�)T⌅µ��⌃�(��T⌅µ⇧)

L = aµ⌥̄⇥µ⌥ + bµ⌥̄⇥5⇥µ⌥ + · · ·

aµ, bµ, . . .

BµBµ = ±b2, b = constant

T⌅µ⇧··· = e a⌅ e b

µ e c⇧ · · · Tabc···

gµ⇧ = e aµ e b

⇧ ⇤ab

�ab ⇤ �a

b + ⇥ab

⇥ab = �⇥ba

xµ ⌅ xµ + ⌃µ

⌃Tabc···⌥ ⇥ tabc··· ⇧= 0

⌃Tabc···⌥ ⇥ tabc··· ⇧= 0

⌃e aµ ⌥ = � a

µ

gµ⇧ = gvacµ⇧ + hµ⇧

gµ⇧ = ⇤µ⇧ + hµ⇧

V = V (T⌥µ�···g⌥�gµ⇥g�⇤ . . . T�⇥⇤··· � t2)

V = V (Tabc···⇧ad⇧be⇧cf . . . T def ··· � t2)

T⌥µ�··· = e �⌥ e ⇥

µ e ⇤� . . . t�⇥⇤···

DµT� = ⌦µT� � �⌥µ�T⌥

(DµT�)2 ⇥ (�⌥µ�t⌥)

2 + · · ·

T� = e a� Ta

(DµT�)2 ⇥ ( b

µ a ⌅a� tb)

2 + · · ·

V ⇤ = ⌃(Bµgµ�B� ± b2) = 0

V ⇤ = ⌃(Bµgµ�B� ± b2) ⌅= 0

bµ(Eµ � 12hµ�b

�) = 0

local frame components spacetime

components e.g.,

also involves the spin connection

appears in covariant derivs. of local tensors

nondynamical in Riemann space (no torsion)

dynamical in Riemann-Cartan space (torsion)

connects spacetime tensors to tensors in local Lorentz frame vierbein

Use a vierbein description in curved spacetime

•  allows spinors (fermions) to be introduced •  gives a structure like a local gauge theory

When is local Lorentz symmetry spontaneously broken?

- rotations & boosts in local frame

- spacetime diffeomorphisms

leave the lagrangian invariant

Tabc··· ⇥ � da � e

b � fc Tdef ··· ⇤ Tabc··· + ⇥ d

a Tdbc··· + ⇥ eb Taec··· + ⇥ f

c Tabf ··· + · · ·

Tabc··· ⇥ Tabc··· + ⇥ da Tdbc··· + ⇥ e

b Taec··· + ⇥ fc Tabf ··· + · · ·

T⇤µ⌅ ⇥ T⇤µ⌅ � (⌃⇤⇧�)T�µ⌅ � (⌃µ⇧�)T⇤�⌅ � (⌃⌅⇧�)T⇤µ� + · · ·� ⇧�(⌃�T⇤µ⌅)

T⇤µ⌅... ⇥ T⇤µ⌅... � (⌃⇤⇧�)T�µ⌅... � (⌃µ⇧�)T⇤�⌅... � · · ·� ⇧�(⌃�T⇤µ⌅...)

T⇤µ⌅... ⇥ T⇤µ⌅... � (⌃⇤⇧�)T�µ⌅... � (⌃µ⇧�)T⇤�⌅... � · · ·

� ba = ⇥ b

a + ⇤ ba

xµ ⇥ xµ + ⌃µ

Tabc··· ⇥ � da � e

b � fc Tdef ··· ⇤ Tabc··· + ⇤ d

a Tdbc··· + ⇤ eb Taec··· + ⇤ f

c Tabf ··· + · · ·

Tabc··· ⇥ Tabc··· + ⇤ da Tdbc··· + ⇤ e

b Taec··· + ⇤ fc Tabf ··· + · · ·

Tabc··· ⇥ Tabc··· + ⇤ da Tdbc··· + ⇤ e

b Taec··· + · · ·

T⌅µ⇧ ⇥ T⌅µ⇧ � (⌥⌅⌃�)T�µ⇧ � (⌥µ⌃�)T⌅�⇧ � (⌥⇧⌃�)T⌅µ� + · · ·� ⌃�(⌥�T⌅µ⇧)

T⌅µ⇧... ⇥ T⌅µ⇧... � (⌥⌅⌃�)T�µ⇧... � (⌥µ⌃�)T⌅�⇧... � · · ·� ⌃�(⌥�T⌅µ⇧...)

T⌅µ⇧... ⇥ T⌅µ⇧... � (⌥⌅⌃�)T�µ⇧... � (⌥µ⌃�)T⌅�⇧... � · · ·

� ba = ⇥ b

a + ⇤ ba

xµ ⇥ xµ + ⌃µ

Tabc··· ⇥ � da � e

b � fc Tdef ··· ⇤ Tabc··· + ⇤ d

a Tdbc··· + ⇤ eb Taec··· + ⇤ f

c Tabf ··· + · · ·

Tabc··· ⇥ Tabc··· + ⇤ da Tdbc··· + ⇤ e

b Taec··· + ⇤ fc Tabf ··· + · · ·

Tabc··· ⇥ Tabc··· + ⇤ da Tdbc··· + ⇤ e

b Taec··· + · · ·

T⌅µ⇧ ⇥ T⌅µ⇧ � (⌥⌅⌃�)T�µ⇧ � (⌥µ⌃�)T⌅�⇧ � (⌥⇧⌃�)T⌅µ� + · · ·� ⌃�(⌥�T⌅µ⇧)

T⌅µ⇧... ⇥ T⌅µ⇧... � (⌥⌅⌃�)T�µ⇧... � (⌥µ⌃�)T⌅�⇧... � · · ·� ⌃�(⌥�T⌅µ⇧...)

T⌅µ⇧... ⇥ T⌅µ⇧... � (⌥⌅⌃�)T�µ⇧... � (⌥µ⌃�)T⌅�⇧... � · · ·

In curved spacetime, the Lagrangian is invariant under both local Lorentz transfs and diffeomorphisms

Lkinetic =1

16⇥GR� 1

4Bµ�B

µ�

Bµ� = DµB� �D�Bµ

Will-Nordvedt

Lkinetic =1

16⇥G

�

a1R + a2BµBµR + a3BµB�Rµ� + a4DµB�D

µB�

+a5DµB�D�Bµ + a6DµBµD�B

� ⇥

L =1

16⇥G

�

a1R + a2BµBµR + a3BµB�Rµ� + a4DµB�D

µB�

+a5DµB�D�Bµ + a6DµBµD�B

� ⇥

� V (BµB� ± b2) + LM

� ba = ⇥ b

a + ⇤ ba

xµ ⇥ xµ + ⌃µ

Tabc··· ⇥ � da � e

b � fc Tdef ··· ⇤ Tabc··· + ⇤ d

a Tdbc··· + ⇤ eb Taec··· + ⇤ f

c Tabf ··· + · · ·

Tabc··· ⇥ Tabc··· + ⇤ da Tdbc··· + ⇤ e

b Taec··· + ⇤ fc Tabf ··· + · · ·

Tabc··· ⇥ Tabc··· + ⇤ da Tdbc··· + ⇤ e

b Taec··· + · · ·

T⌅µ⇧ ⇥ T⌅µ⇧ � (⌥⌅⌃�)T�µ⇧ � (⌥µ⌃�)T⌅�⇧ � (⌥⇧⌃�)T⌅µ� + · · ·� ⌃�(⌥�T⌅µ⇧)

T⌅µ⇧... ⇥ T⌅µ⇧... � (⌥⌅⌃�)T�µ⇧... � (⌥µ⌃�)T⌅�⇧... � · · ·� ⌃�(⌥�T⌅µ⇧...)

T⌅µ⇧... ⇥ T⌅µ⇧... � (⌥⌅⌃�)T�µ⇧... � (⌥µ⌃�)T⌅�⇧... � · · ·

DµJµ = 0

Lint = BµJµ

MPlanck =�

h̄c/G ⇤ 1019 GeV

MPlanck =

⌅⇤⇤⇤⇤⇤⇤⇥

h̄c

G⇤ 1019 GeV

Bµ = bµ + Aµ

⇧T ⌃ ⌅= 0

L � ⇧Mk ⇧T ⌃� �̄(i )k⌥

[xµ, x⌃] = i⌅µ⌃

L � 1

4iq ⌅�⇥ F�⇥ �̄ ⇤µ Dµ�

L⇥L

vacuum breaks Lorentz symmetry

get fixed background tensors in local frames

Local SLV occurs when a local tensor has a nonzero vev

can introduce a tensor vev using a potential V

has a minimum for a nonzero local vev

where

quadratic potential

In gauge theory SSB has well known consequences:

(1) Goldstone Thm: when a global continuous sym is spontaneously broken massless Nambu-Goldstone (NG) modes appear

(2) Higgs mechanism: if the symmetry is local the NG modes can give rise to massive gauge-boson modes.

e.g. W,Z bosons acquire mass

e.g. Higgs boson

(3) Higgs modes: depending on the shape of the potential, additional massive modes can appear as well

With SSB the theory has multiple potential vacuum solutions

NG excitations stay inside the potential minimum obey V’ = 0

Massive Higgs modes climb up the potential walls

obey V’ ≠ 0

V’ = 0 in the minimum

A vacuum solution is Spontaneously chosen

If NG modes exist, they might possibly be: known particles (photons, gravitons) noninteracting or auxiliary modes gauged into gravitational sector (modified gravity) “eaten” (Higgs mechanism)

Can use models with SLV to address these questions:

•  Bumblebee models

•  Cardinal models

•  Antisymmetric two-tensor models

Bµ� Aµ

Cµ� � gµ�

Hµ�

Bµ� Aµ

Cµ� � gµ�

Hµ�

photons?

gravitons?

Question: Can NG modes or a Higgs mechanism occur if Lorentz symmetry is spontaneously broken?

Consider a theory with a tensor vev in a local Lorentz frame:

spontaneously breaks local Lorentz symmetry

The vacuum vierbein is also a constant or fixed function

e.g., assume a background Minkowski space with

The spacetime tensor therefore also has a vev:

vierbein vev

spontaneously breaks diffeomorphisms

Spontaneous breaking of local Lorentz symmetry implies spontaneous breaking of diffeomorphisms

III. Nambu-Goldstone Modes & Higgs Mech.

How many NG (or would-be NG) modes can there be?

6 broken Lorentz generators 4 broken diffeomorphisms

Can have up to

There are potentially 10 NG modes when Lorentz symmetry is spontaneously broken

Where are they? answer in general is gauge dependent

But for one choice of gauge can put them all in the vierbein

No Lorentz SSB has 16 components - 6 Lorentz degrees of freedom - 4 diff degrees of freedom

up to 6 gravity modes (GR has only 2)

With Lorentz SSB all 16 modes can potentially propagate

Perturbative analysis:

can drop distinction between local & spacetime indices

Small fluctuations

10 symmetric comps.

6 antisymmetric comps.

in general there are many such possible excitations

Vacuum

NG Modes: The NG modes are the excitations from the vacuum generated by the broken generators that maintain the extremum of the action:

where

Note: condition also follows from an SSB potential of form

minimum of V <T> = t

This condition is satisfied by:

the vierbein contains the NG excitations

Lorentz & diffeo NG excitations maintain tensor magnitudes

NG excitations:

The combination contains the NG degrees of freedom

Expand the vierbein to identify the NG modes

Can find an effective theory for the NG modes by performing small virtual particle transformations from the vacuum and promoting the excitations to fields.

Under LLTs:

Under diffs:

Promote the NG excitations to fields:

write down an effective theory for them

(leading order)

Results: we find that the propagation & interactions of the NG modes depends on a number of factors:

•  Geometry

•  VEV

•  Ghosts

-  Minkowski -  Riemann -  Riemann-Cartan

- constant vs. nonconstant <T>

- kinetic terms with ghost modes permit propagation of additional NG modes

How many NG modes there are in a given theory will in general depend on all these quantities

As an example, will consider a vector model in Riemann spacetime and in Riemann-Cartan spacetime.

Can a Higgs mechanisms occur?

there are 2 types of NG modes (Lorentz & diffs) therefore have potentially 2 types of Higgs mechanisms

diffeomorphism modes:

connection depends on derivatives of the metric no mass term for the vierbein (or metric) itself

conventional mass term

can a Higgs mechanism occur for the diffs? does the vierbein (or metric) acquire a mass?

No conventional Higgs mechanism for the metric (no mass term generated by covariant derivatives)

but propagation of gravitational radiation is affected

Lorentz modes:

go to local frame (using vierbein)

Get quadratic mass terms for the spin connection

gauge fields of Lorentz symmetry

suggests a Higgs mechanism is possible for ωµab

only works with dynamical torsion allowing propagation of ωµab

Lorentz Higgs mechanism only in Riemann-Cartan spacetime

offers new possibilities for model building theories with dynamical propagating spin connection finding models with no ghosts or tachyons is challenging

Are there additional massive Higgs modes?

•  consider excitations away from the potential minimum

unconventional mass term

different from nonabelian gauge theory (no Aµ in V) here the gauge field (metric) enters in V

metric and tensor combine as additional massive modes

Expand

SLV can give rise to massive Higgs modes involving the metric

Find mass terms for combination of and

appear as excitations with

Note: BB models do not have local U(1) gauge invariance (destroyed by presence of the potential V)

vector field

Potential

Vev

Gravity theories with a vector field and a potential term that induces spontaneous Lorentz breaking

Bumblebees: theoretically cannot fly (and yet they do)

First restrict to Riemann spacetime (no torsion) no Higgs mechanism for Lorentz NG modes

IV. Example: Bumblebee Models

Will then look at possibility of a Higgs Mechanism

Bumblebee Lagrangian:

depending on the interpretation of the vector

Have different choices for the kinetic, potential, & int terms

minimum of V gives the vev

vector in a vector-tensor theory of gravity

set gravitational couplings only

generalized vector potential (photons?)

keep allows Lorentz violating matter ints

For

Or for

Bµ

LB = �1

4Bµ�B

µ�

Lint

Lint = 0

Lint ⇧= 0

RPlanck ⇤ 10�35 m

L = L0 � V + Lint

⇥0V⌅ = 0

⇥0V⌅ ⇤ 0

Jµ ⇥ 0

Jµ = 0

Bµ

LB = �1

4Bµ�B

µ�

Lint

Lint = 0

Lint ⇧= 0

RPlanck ⇤ 10�35 m

L = L0 � V + Lint

⇥0V⌅ = 0

⇥0V⌅ ⇤ 0

Jµ ⇥ 0

Jµ = 0

Bµ

LB = �1

4Bµ�B

µ�

Lint

Lint = 0

Lint ⇧= 0

RPlanck ⇤ 10�35 m

L = L0 � V + Lint

⇥0V⌅ = 0

⇥0V⌅ ⇤ 0

Jµ ⇥ 0

Jµ = 0

Bµ

LB = �1

4Bµ�B

µ�

Lint

Lint = 0

Lint ⇧= 0

RPlanck ⇤ 10�35 m

L = L0 � V + Lint

⇥0V⌅ = 0

⇥0V⌅ ⇤ 0

Jµ ⇥ 0

Jµ = 0

LB =1

16⇥G(R�2�)+⇤1B

µB�Rµ�+⇤2BµBµR�1

4⌅1Bµ�B

µ�+1

2⌅2DµB�D

µB�+1

2⌅3DµBµD�B

��V (BµBµ⇥b2)+LM

L =1

16⇥G(R� 2�) + LB � V (BµBµ ⇥ b2) + LM

LB = +⇤1BµB�Rµ� + ⇤2B

µBµR� 1

4⌅1Bµ�B

µ� +1

2⌅2DµB�D

µB� +1

2⌅3DµBµD�B

�

LB = +⇤1BµB�Rµ� + ⇤2B

µBµR

�1

4⌅1Bµ�B

µ� +1

2⌅2DµB�D

µB� +1

2⌅3DµBµD�B

�

L =1

16⇥G(R� 2�) + LB � V (BµBµ ± b2) + Lint

LB =1

16⇥GR� 1

4Bµ�B

µ�

Bumblebee Kinetic Terms:

(1)   Bµ as in a vector-tensor theory of gravity

(2) Bµ as a generalized vector potential

models with Will-Nordvedt kinetic terms

Kostelecky-Samuel models

expect propagating ghost modes

no propagating ghost modes

charged matter interactions

global U(1) charge with

Bµ

LB = �1

4Bµ�B

µ�

Lint

Lint = 0

Lint ⇧= 0

RPlanck ⇤ 10�35 m

L = L0 � V + Lint

⇥0V⌅ = 0

⇥0V⌅ ⇤ 0

Jµ ⇥ 0

Jµ = 0

LB =1

16⇥G(R�2�)+⇤1B

µB�Rµ�+⇤2BµBµR�1

4⌅1Bµ�B

µ�+1

2⌅2DµB�D

µB�+1

2⌅3DµBµD�B

��V (BµBµ⇥b2)+LM

L =1

16⇥G(R� 2�) + LB � V (BµBµ ⇥ b2) + LM

LB = +⇤1BµB�Rµ� + ⇤2B

µBµR� 1

4⌅1Bµ�B

µ� +1

2⌅2DµB�D

µB� +1

2⌅3DµBµD�B

�

LB = +⇤1BµB�Rµ� + ⇤2B

µBµR

�1

4⌅1Bµ�B

µ� +1

2⌅2DµB�D

µB� +1

2⌅3DµBµD�B

�

L =1

16⇥G(R� 2�) + LB � V (BµBµ ± b2) + Lint

LB =1

16⇥GR� 1

4Bµ�B

µ�

Bumblebee Potential Terms:

(1)   Lagrange-multiplier potential

(2) Smooth quadratic potential

allows massive-mode field no Lagrange multiplier

freezes out massive mode appears as an extra field

Both exclude local U(1) symmetry

NG & massive modes: Examine different types of bumblebee models to look at the:

degrees of freedom behavior of NG & massive modes

Are the models stable (positive Hamiltonian)?

e.g., flat spacetime with a timelike vev

⇒  initial values with exist ⇒  ultimately means bumblebee models are useful at low energy as effective or approx theories

L = LB + V (BµBµ ± b2)

�µ =�L

�(⇥0Bµ)

H = �µ⇥0Bµ � L

H > 0

H < 0

L = LB + V (BµBµ ± b2)

�µ =�L

�(⇥0Bµ)

H = �µ⇥0Bµ � L

H > 0

H < 0

bµ = (b, 0, 0, 0)

can perform a Hamiltonian constraint analysis

KS models

L = LB + V (BµBµ ± b2)

�µ =�L

�(⇥0Bµ)

H = �µ⇥0Bµ � L

H > 0

H < 0

⇒  can find subspace of phase space with •  λ = 0 (Lagrange-multiplier V) •  large mass limit (quadratic V)

⇒  in these subspaces, the KS model matches EM

field strength

quadratic potential

matter current

Example: KS Bumblebee model in Riemann spacetime

timelike vev

Expect up to 4 massless NG modes what are they? do they propagate?

Theory can have a massive mode how does it affect gravity?

No conventional Higgs mechanism Riemannn spacetime

Equations of motion:

where

massive mode obeys

NG modes alone obey Einstein-Maxwell eqs

massive mode acts as source of charge & energy has nonlinear couplings to gravity and Bµ equations can’t be solved analytically

To illustrate the behavior of the NG & massive modes, it suffices to work with linearized equations of motion

the massive mode acts as a static primordial charge density that does not couple with matter current Jµ

static massive mode get that

linearized theory is stable in flat-spacetime limit massive mode acts as source of charge & energy equations can be solved

With global U(1) matter couplings

can restrict to initial values that stabilize Hamiltonian conservation of conventional matter charge holds massive mode charge density decouples

Find that the diff NG mode drops out of

the diff NG mode does not propagate it is purely an auxiliary field

obey axial gauge condition

Lorentz NG modes are two transverse massless modes propagate as photons in axial gauge (linearized theory)

Find that the Lorentz NG modes propagate

Lorentz NG excitations

Fate of NG modes

and

removes massive mode from propagating degrees of freedom

has a nonzero vev for the EM field classically equivalent to electromagnetism

§  Nambu (1968) - local U(1) vector theory in nonlinear gauge

Idea of photons as NG modes

collective fermion excitations give rise to composite photons emerging as NG modes

§  Bjorken (1963) – composite fermion models

Neither gives signals of physical Lorentz violation

has no local U(1) gauge invariance NG modes behave like photons has signatures of physical Lorentz violation includes gravity (local Lorentz symmetry)

Here the KS bumblebee model is different

Can the Einstein-Maxwell solutions originate out of a theory with spontaneous Lorentz violation but no local U(1) symmetry?

To answer this, must look at effects of the massive mode

models with massive modes are not equiv to EM

Consider a point mass m with charge q in weak static limit

usual potentials

Introduce a potential for the massive mode

modifies EM and gravitational fields

modified Newtonian potential

Special cases:

(i) no charge couplings

and decouple from matter

Newton’s constant rescales

purely modified gravity (no electromagnetism) NG modes not photons (what are they?)

e.g., with

Attempt to fit to yield a suitable form of

that describes a modified theory of gravity

models of dark matter? modified Newtonian potential (altered 1/r dependence)

There are numerous examples that could be considered

(ii) no massive mode

and usual electromagnetic fields

large mass limit same solutions emerge with a massive mode when

usual Newtonian potential

clearly the most natural choice

The Einstein-Maxwell solution (with two massless transverse photons and the usual static potentials) emerges from the KS bumblebee with spontaneous Lorentz breaking but no local U(1) gauge symmetry

matter interactions with bµ signal physical Lorentz breaking

(iii) heavy massive mode

Higgs Mechanism Riemann-Cartan Spacetime:

dynamical spin connection

and (tetrad postulate)

To quadratic order, the kinetic term becomes

quadratic “mass” terms in ωµab

Suggests a Higgs mechanism is possible for ωµab

Note: Only works in the context of a theory with dynamical torsion allowing propagation of ωµ

ab

Can get a Higgs mechanism in Riemann-Cartan spacetime

and

Model Building in Riemann-Cartan Spacetime:

consider propagating ωµab in a flat background

need to add a kinetic term for ωµab

Ghost-free models are extremely limited

the massless modes must match with

Results for ghost-free models: models with propagating massless ωµ

ab exist e.g.,

but it is very hard to find a straightforward ghost-free Higgs mechanism for the spin connection

it remains an open problem

Tensor Models

symmetric 2-tensor Cµν in Minkowski space with SLV Cardinal Model

NG modes obey linearized Einstein eqs in fixed gauge nonlinear theory generated using a bootstrap mechanism

alternate theory of gravity that contains GR at low energy

anti-symmetric 2-tensor Bµν coupled to gravity with SLV Phon Model

up to 4 NG modes called phon modes (phonene) certain models produce a scalar (inflaton scenarios)

massive modes exist that can modify gravity

In gravity models with spontaneous Lorentz breaking diffeomorphisms also spontaneously broken both NG and massive modes can appear

possibility of a Higgs mech. for spin connection

-Riemann spacetime: no conventional Higgs mech. for the metric but massive Higgs modes can involve the metric massive modes can affect the Newtonian potential

-Riemann-Cartan spacetime: Gravitational Higgs effect depends on the geometry

V. Conclusions

Bumblebee Models NG modes propagate like massless photons massive mode modifies Newtonian potential Einstein-Maxwell solution is special case

Open Issues & Questions

è must eliminate ghosts è quantization è Higgs mechanism with massive spin connection è photon models with signatures of SLV

Physically viable models with SLV?

SME with gravity

è role of NG modes in gravitational sector? è massive Higgs modes? è origin of SME coefficients?

Primary References: Kostelecky & Samuel, PRD 40 (1989) 1886 Kostelecky, PRD 69 (2004) 105009 RB & Kostelecky, PRD 71 (2005) 065008 RB, Fung & Kostelecky, PRD 77 (2008) 065020 RB, Gagne, Potting, & Vrublevskis, PRD 77 (2008) 125007