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
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