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

Lorentz symmetry comes in two varieties:

global ⇒ symmetry of - field theories invariant under global LTs

local ⇒ symmetry of - Lorentz symmetry holds locally

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

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

 includes gravity, SM, and LV sectors

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

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

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

⇒ local Lorentz frame components Tabc. . . 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 appears in cov. derivs. spin connection: ⇒ of local tensors

In Riemann spacetime with (metric)

⇒ spin connection is determined by the vierbein ⇒ not independent degrees of freedom Can also introduce torsion Tλµν = Γλµν - Γλνµ ⇒ spin connection becomes dynamically independent ⇒ gives gravity the form of a

→ 16 components

→ 24 components

New geometry emerges:

Riemann-Cartan curvature = Rκλµν ⇒ spacetime torsion = Tλµν

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

The SME with gravity includes curvature & torsion Constructing the SME with Gravity

Example: fermion coupled to gravity:

where

Additional fermion couplings might include: 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)

Riemannian limit (zero torsion):

Jay Tasson’s talk will look at phenomenology 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

But with spontaneous LV ⇒ arise as vev’s ⇒ must be treated dynamically

No-go theorem:  explicit breaking incompatible with geometrical identities, but spontaneous symmetry breaking evades this difficulty Spontaneous Lorentz Violation (SLV)

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

originally motivated from quantum gravity &

Open Problem General Relativity is a classical theory  not compatible with quantum physics

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

Planck scale:

Is Lorentz symmetry exact at the Planck scale? String Theory & SLV

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

• Nonpertubative vacuum in string field theory • Produces vevs for tensor fields <Τ> ≠ 0

⇒ can lead to spontaneous Lorentz violation

⇒ provides most elegant form of Lorentz violation • fundamental theory fully Lorentz invariant • vacuum breaks Lorentz symmetry • evades the no-go theorem

SME coeffs., e.g., aµ, bµ, cµν, dµν, Hµν, . . . arise as vacuum expectation values when SLV occurs II. Spontaneous Lorentz Violation 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 Spontaneous symmetry breaking occurs in gauge theories

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

a potential V has a nonzero minimum

V The theory has multiple potential vacuum solutions

 the physical vacuum picks one, breaking the symmetry f 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

const. scalar field (electroweak) V ≠ 0

tensor vev ≠ 0 f vacuum breaks Lorentz symmetry 1 eµ⌥ = ⇧µ⌥ +(2hµ⌥ + µ⌥)

hµ⌥ = h⌥µ

µ⌥ = ⌥µ 1 1 µ Tµ⌥ tµ⌥ kinetic = R BµB ⇤ ···⌅⇥ ··· L 16⇥G 4 1 eµ⌥ = ⇧µ⌥ +( hµ⌥ + µ⌥) ⌅T⌃µ⌥ 2 =(T⌃µ⌥ t⌃µ⌥ ) Bµ = DµB DBµ ··· ··· ··· Will-Nordvedt hµ⌥ = h⌥µa b c T⌅µ⌃⇧µ⌥ ==(e⌅ Te⌃µµ⌥e⇧ t⌃Tµabc⌥ ) ······ ··· ···1 ······ µ µ µ kinetic = a1R + a2BµB R + a3B B Rµ + a4DµBD B ⌃ µµ⌥⌥ = L⌥µ a b 16⇥⇤⇥G 2 T ···gg⌃µ⇧g=µ⇥egµ⌥⇤e...T⇧ ⇤ab ··· = t µ µ Bµ+=a5bDµ µ+BADµ B + a6DµB DB ⇥ How is SLV introduced? 2Tµ⌥ ⌃µ⌥a tµ⌥ a a1 ⇥⇤ ⇤t = ···t ⌅⇥ ···b ⇧⌃···⇧µb ⇥+⇧⌥⇤⇥ b...t ··· µ µ µ Consider a Lorentz-invariant lagrangian =with tensora1R fields+ a2B µTB =0R + a3B B Rµ + a4DµBD B Bµ = bµ + Aµ ⇤ LBµ =16⇥bµG+ Aµ ⌃ ⌥⇧ ⌅T⌃µ⌥ =(⌃Tµ⌃⌥µ⌥ t⌃µ⌥ ) ⇥⇤ µ 2 µ 2 include a potential V =thatV ( Thas a··· ⇥nontrivialg =g +ga⇥⌥⇤ Dminimum...TB D B ··· +⌃ta)D B D Bk V (B B b )+ ··· ···ab⌃ µ⇥ 5ba···µ 6 Tµ ⌦¯(i↵) ⇥ µ M that occurs when T has=0 a nonzero vev T =0 Mk ± L ⌃ ⌥⇧ ⌃ ⌥⇧ L ⇥ ⌃ ⌥ ⌥µ⌃µ⌥ =(T⌃µ⌥µ ⇥⇤t⌃µµ⇥⌥ ⇤)µ2 e.g., Vin =flatV (spacetime,T ···g⌥···g Tµwith⌃⇥µg⌥⇤ componentsx...T=···e⌃ xeµ···+e ···⌥⌃Tt...t) ⇥⇤ dµ e⌥ f T µ⌥ ⌃ T ⌦¯(i↵)···k ⌅ ⌃ Tabc ⌦¯(i↵···)ka [x b,x c]=defi⇧ Mk Mk ··· ⇥ ··· L ⇥ T ⌃µ⌃⌥ ⌥g g g ...TL ⇥⇥⇤ ⌃=⌥t2d e f V = V (T abc ···⇧ ⌃⇧ µ⇧⇥ T⌥⇤abc...TdefTabctabc··· t+2=0)⇤a Tdbc 1 + ⇤b Taec + ⇤c Tabf ··· ad be ⌃cf ⇤⌥⇥··· ··· ⇧ ··· ⇥ ··· ¯ µ ··· [xµ,x⌥]=i⇧µ⌥ ··· [xµ,x···⌥]=i⇧µ⌥ iq ⇧ F⇥ ⌦⇤ Dµ ⌦ 2 ⌃µ⌥ abc abc⇥⇤ L ⇥ 4  t = t ···⇧⇥ T⇧⇤ ⇧T⌥⇤ ...tt T =0··· ( ⌃ )Tµ⇧ (µ⌃ )T (⇧⌃ )T ⌃ (T ) has a minimumT = whene e ⌃e µ...t⇥··· ⌅µ⇧ ⇥···⌅µ⇧ ⌅ ⌅⇧ ⌅µ ⌅µ⇧ 1⌥µ ⇥ ⌥ µ µ⌃ ⌥⇥⇥⇤1 ⇥ ⇧ µ iq ⇧··· F ⌦⇤¯ Dµ ⌦ iq···⇧ F ⌦⇤¯ Dµ ⌦ L ⇥ 4 2 ⇥abc L ⇥a4 defa ⇥ ¯ µ L¯ ⇤Lµ where t = t ···⇧ad⇧be⌥⇧ecfµ ...t= µ=···aµ⌥⇥ ⌥ + bµ⌥⇥5⇥ ⌥ + DµT = ⌦µT ⌃µT⌥⌥ L ··· = gravity + SM + LV + What about in curved ⌃spacetime?µ⌥ vac ⇥⇤ L 2L L L ··· V = VL(T⇤2L···g⌃⌥ggµµ⇥⇧g2⌥⇤= ...Tgµ⇧ +Lh⇤µ···⇧L t )aµ,bµ,... (DµT) (µt⌥) + b b Lorentz= symmetry+ ⇥ +is a local=+ symmetry··· + +Ta ⇥+a Tb Ta + ⌅a Tb L Lgravity LSM LLVL L···gravity LSMµ LLV⇤2 ··· ⌅ a gµ⇧ = ⇤µ⇧ +BhµµB⇧ = b ,b= constant T = e Ta ± b b b Tµ Tµb (↵µ )T (↵Tµ) Ta ⇥ T Ta + ⌅ TTa ⇥ T Ta + ⌅ T ⇤ a b ⌅ a b ⇤ a b ⌅ ⇤ a b (D T )2 ( b ⌅a t )2 + µ µa b Tµ Tµ (↵µ ⇥)T Tµ(↵TTµµ) ···(↵µ )T (↵Tµ) ⇤ ⇤ µ 2 V = ⌃(Bµg B b )=0 ⇤ ± µ 2 V = ⌃(Bµg B b ) =0 ⇤ ± ⌅ µ 1 b ( µ hµb )=0 E 2 Use a vierbein description in curved spacetime

vierbein connects spacetime tensors  to tensors in local Lorentz frame

local frame spacetime components components e.g.,

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

 also involves the spin connection

appears in covariant derivs.  of local tensors

nondynamical in Riemann space (no torsion) dynamical in Riemann-Cartan space (torsion) 1 1 µ kinetic = R BµB µ L 16⇥G 4 DµJ =0

µ Bµ = DµB DBµ int = BµJ Will-Nordvedt L 1 19 µ µ µ b bMPlanckb = hc/G¯ 10 GeV = a R + a BµB R + a B B Rµ + a DµBD B a = ⇥a + ⇤a Lkinetic 16⇤⇥G 1 2 3 4 µ µ ⌅hc¯ ⇤ 19 +a D B D B + a D B D B µ µ µ ⇤ 5 µ 6 µ ⇥ M = ⇤ 10 GeV x x + ⌃Planck ⇤ ⇤ ⇥ ⇤ G ⇤ ⇥ 1 d e f d e f µ µ µ In curved spacetime, the Lagrangian = is invarianta R + a BµB R + a B B Rµ + a DµBD B T Tabc da eb f Tc Tdef T Tabc ++⇤ ⇥daT Tdbc +L+⇤ ⇥ebT16Taec⇥G ++1⇤⇥fc TTabf2 ++ 3 4 abc ··· ⇥undera b bothc def local ···Lorentz⇤abc ···transfsa dbc and···B diffeomorphismsµ =b bµaec+ A···µ c abf ··· ··· ··· ⇥ ··· ⇤ ··· ··· ··· µ ··· ···µ 2 d e +a5Df µBD B + a6DµB DB ⇥ V (BµB b )+ M Tabc Tabc + d⇥a Tdbc + ⇥eb Taec +b⇥fc Tabfb +b ± L Tabc- rotations··· ⇥Tabc & +···boosts⇤a Tdbc in local···+ ⇤ bframeTaec ···+aT⇤c ==0T⇥abfa +···+⇤a ··· ··· ⇥ ··· ··· ··· ⇧ ⌃⌅ ··· ··· T⇤µ⌅ T⇤µ⌅ (⌃⇤⇧ )Tµ⌅ (⌃µ⇧d )T⇤⌅ (b⌃e=⌅⇧⇧⇥µ)TbT+⇤µ⇤µ¯b+(i )kµ⌥ ⇧ (⌃T⇤µ⌅) ⇥ Tabc Tabc + ⇤a Tdbc +⇤ab MTkaecxa +xa +···⌃ ··· ⇥ ··· ···L ⇧ ···⇥⌃ ··· T⇤µ⌅... - spacetimeT⇤µ⌅... (diffeomorphisms⌃⇤⇧ )Tdµ⌅e... f (⌃ µ ⇧ x)µT⇤⌅µ x...⌃µ + ⌃µ µd⌃ ⇧ (⌃T⇤µe⌅...) f T T ⇥ (⌥ ⌃Tabc)Tµ⇧ (a⌥µ⌃b )TcTdef (⌥[⇧x⌃T,xabc)T ]=+···i+⇤⌅aTdbc ⌃+(⌥⇤bTTaec ) + ⇤c Tabf + ⌅µ⇧ ⇥ ⌅µ⇧ ⌅ ··· ⇥ ⌅⇧ ··· ⇤⇥ ···⌅µ ······ ⌅µ⇧··· ··· ··· d e f 1 ⇥d µ e f TT⇤abcµ⌅... T⇤aµ⌅...bTc(⌃T⇤⇧def)Tµ⌅T...abc+iq⇤(⌃d⌅+µT⇧⇤aF)TT⇤⌅dbc+⇤¯⇤...e+TD⇤b Taec+ ⇤ f+T⇤c Tabf+ + T T ⇥⇥(⌥ ⌃ )Tabcµ⇧... (⌥abcµ⇤⌃ )T a dbc ⇥ b⌃aec(⌥···µT ···c ) abf ··· ⌅µ⇧... ⇥ ⌅µ⇧...··· ⌅ ··· ⇥ ···L ··· 4⌅⇧··· ... ········· ··· ⌅µ⇧... ··· ······ d e f leave theTabc lagrangianTabcT invariant+ ⇤a TTdbc ++ ⇤⇤bdTaec ++ ⇤⇤ceTTabf ++ T T ··· (⇥⌥ ⌃ )abcT···µ⇧... abc(⌥µ···⌃ )Ta dbc··· b aec ··· ··· ⌅µ⇧... ⇥ ⌅µ⇧... ⌅ ··· ⇥ ···L ⇥⌅⇧L ... ··· ··· ··· ··· d e T⌅µ⇧ T⌅Tµ⇧abc (⌥⌅⌃Tabc)Tµ⇧+ ⇤a(⌥Tµdbc⌃ )T+⌅⇧⇤b Taec(⌥⇧⌃ +)T⌅µ + ⌃ (⌥T⌅µ⇧) When is local⇥ Lorentz ···symmetry⇥ ··· spontaneously ··· broken? ··· ··· ··· T⌅µ⇧ T⌅µ⇧ (⌥⌅T⌃ )Tµ⇧ (⌥ (⌃⌥µ)⌃T)µT⇧⌅⇧... (⌥(⌥µ⇧⌃⌃ )T)T⌅µ + ⌃⌃ ((⌥⌥TT⌅µ⇧) ) ⇥ ⌅µ⇧...⇥ ⌅µ⇧... ⌅ ⌅⇧... ······ ⌅µ⇧... T⌅µ⇧... T⌅µT⇧... (⌥⌅⌃T)Tµ⇧... (⌥ (⌃⌥µ)⌃T)µT⇧⌅⇧...... (⌥µ⌃ )T ⌃ (⌥T⌅µ⇧...) ⇥ ⌅µ⇧... ⇥ ⌅µ⇧... ⌅ ···⌅⇧... ··· T T (⌥ ⌃ )Tµ⇧... (⌥µ⌃ )T ⌅µ⇧... ⇥ ⌅µ⇧... ⌅ ⌅⇧... ··· Local SLV occurs when a local tensor has a nonzero vev

vacuum breaks Lorentz symmetry

get fixed background tensors in local frames

can introduce a tensor vev using a potential V

quadratic potential

 has a minimum for a nonzero local vev

where 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) : if the symmetry is local the NG modes can give rise to massive gauge-boson modes.

e.g. W,Z bosons acquire mass

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

e.g. Higgs boson With SSB the theory has multiple potential vacuum solutions

V’ = 0 in the minimum

A vacuum solution is Spontaneously chosen

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

Massive Higgs modes  climb up the potential walls

obey V’ ≠ 0 Question: Can NG modes or a Higgs mechanism occur if Lorentz symmetry is spontaneously broken?

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

Can use models with SLVB µto addressAµ these questions: • Bumblebee models Bµ Aµ photons? Cµ gµ • Cardinal models gravitons? Cµ gµ • Antisymmetric two-tensor models Hµ Hµ III. Nambu-Goldstone Modes & Higgs Mech. 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

vierbein vev The spacetime tensor therefore also has a vev:

spontaneously breaks diffeomorphisms

Spontaneous breaking of local Lorentz symmetry implies spontaneous breaking of diffeomorphisms How many NG (or would-be NG) modes can there be? Can have up to 6 broken Lorentz generators 4 broken diffeomorphisms

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: Small fluctuations can drop distinction between local & spacetime indices

10 symmetric comps.

6 antisymmetric comps.

Vacuum

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

in general there are many such possible excitations Lorentz & diffeo NG excitations maintain tensor magnitudes

where

Note: condition also follows from an SSB potential of form

minimum of V  = t

This condition is satisfied by:

the vierbein contains the NG excitations Expand the vierbein to identify the NG modes

NG excitations:

The combination contains the NG degrees of freedom

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: (leading order)

Under diffs:

Promote the NG excitations to fields:

write down an effective theory for them Results: we find that the propagation & interactions of the NG modes depends on a number of factors:

• Geometry - Minkowski - Riemann - Riemann-Cartan

• VEV - constant vs. nonconstant

• Ghosts - 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: can a Higgs mechanism occur for the diffs? does the vierbein (or metric) acquire a mass?

conventional  mass term

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

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)

gauge fields of Lorentz symmetry Get quadratic mass terms for the spin connection

ab suggests a Higgs mechanism is possible for ωµ ab only works with dynamical torsion allowing propagation of ωµ

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 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

Find mass terms for combination of and

appear as excitations with

SLV can give rise to massive Higgs modes involving the metric IV. Example: Bumblebee Models Gravity theories with a vector field and a potential term that induces spontaneous Lorentz breaking

vector field  Potential  Vev 

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

Bumblebees: theoretically cannot fly (and yet they do)

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

Will then look at possibility of a Higgs Mechanism 1 µ µ 1 µ 1 µ 1 µ µ 2 = (R 2)+⇤ B B Rµ +⇤ B BµR ⌅ BµB + ⌅ DµBD B + ⌅ DµB DB V (BµB b )+ LB 16⇥G 1 2 4 1 2 2 2 3 ⇥ LM

1 µ 2 = (R 2)+ V (BµB b )+ L 16⇥G LB ⇥ LM

µ µ 1 µ 1 µ 1 µ =+⇤ B B Rµ + ⇤ B BµR ⌅ BµB + ⌅ DµBD B + ⌅ DµB DB LB 1 2 4 1 2 2 2 3

µ µ =+⇤ B B Rµ + ⇤ B BµR LB 1 2 1 µ 1 µ 1 µ ⌅ BµB + ⌅ DµBD B + ⌅ DµB DB Bumblebee Lagrangian:4 1 2 2 2 3 1 µ 2 = (R 2)+ V (BµB b )+ L 16⇥G LB ± Lint

1 1 µ minimum =of V givesR theBµ Bvev LB 16⇥G 4 B Have different choicesµ for the kinetic, potential, & int terms 1 µ depending= Bon Btheµ Binterpretation of the vector LB 4µ 1 µ For Bµ B=vectorBµ inB a vector-tensor theory of gravity L L4int 1 µ B = setB µB int =0 gravitational couplings only L 4 LLint =0 Or for B Lgeneralizedintint=0⇧ vector potential (photons?) Lintµ L R 10 35 m keep1 Planck µint =0 allows Lorentz violating matter ints = int =0BµBL ⇤⇧ LB L4 R = 0 10V +35 mint int L=0PlanckL ⇤ L L int⇧ L = 35⇥0V ⌅V=0+ RPlanck L10L0 m Lint int⇤=0 L ⇥ V 0 = V +⇥00V ⌅⌅=0⇤ L L0 Lint int =0 µ L ⇧ ⇥JV 0 ⇥0V ⌅ =0 0 ⌅⇥ R 10 35 m ⇤ Planck Jµ =0 ⇤ Jµ 0 ⇥0V ⌅ 0 ⇥ = 0 V⇤ + int L L LJµ =0 Jµ 0 ⇥0V ⇥⌅ =0 Jµ =0 ⇥ V 0 0 ⌅ ⇤ Jµ 0 ⇥ Jµ =0 1 µ µ 1 µ 1 µ 1 µ µ 2 = (R 2)+⇤ B B Rµ +⇤ B BµR ⌅ BµB + ⌅ DµBD B + ⌅ DµB DB V (BµB b )+ LB 16⇥G 1 2 4 1 2 2 2 3 ⇥ LM

1 µ 2 = (R 2)+ B V (BµB b )+ M BumblebeeL 16⇥ GKinetic Terms:L ⇥ L

µ (1) B as inµ a vector-tensor1 µ theory1 of gravityµ 1 µ B =+⇤1B B Rµ +µ ⇤2B BµR ⌅1BµB + ⌅2DµBD B + ⌅3DµB DB L models with4 Will-Nordvedt2 kinetic terms2 µ µ =+⇤ B B Rµ + ⇤ B BµR LB 1 2 1 µ 1 µ 1 µ ⌅ BµB + ⌅ DµBD B + ⌅ DµB DB 4 1 2 2 2 3 1 expect propagatingµ 2 ghost modes = (R 2)+ V (BµB b )+ L 16⇥G LB ± Lint (2) Bµ as a generalized vector potential 1 1 µ  = RB BµB Kostelecky-SamuelLB 16⇥G µ4 models 1 µ no propagating = BµB  LB 4 ghost modes

charged matter interactions Lint with =0global U(1) charge Lint =0 Lint ⇧ R 10 35 m Planck ⇤ = V + L L0 Lint

⇥0V ⌅ =0

⇥ V 0 0 ⌅ ⇤ Jµ 0 ⇥ Jµ =0 Bumblebee Potential Terms:

(1) Lagrange-multiplier potential

freezes out massive mode appears as an extra field

(2) Smooth quadratic potential

allows massive-mode field no Lagrange multiplier

Both exclude local U(1) symmetry µ 2 = + V (BµB b ) L LB ± µ = L µ 2 NG & massive modes:= + V (BµB b ) (⇥0Bµ) L LB ± Examine different types of bumblebee models to look at the: µ µ = ⇥ Bµ degrees of freedom = L 0 (⇥ B ) H L behavior of NG & massive0 µ modes > 0 Are the models stable (positiveµ Hamiltonian)? H = ⇥0Bµ can performH a Hamiltonian L constraint analysis< 0 µ 2 = + HV (BµB b ) > 0 L LB ± e.g., flat spacetime withH a timelike vev bµ =(b, 0, 0, 0) µ ⇒ initial values with < 0 exist = L H (⇥0Bµ) ⇒ ultimately means bumblebee models are useful at low energy as effective or approx theoriesµ = ⇥0Bµ KS models H L ⇒ can find subspace of phase space with > 0 H • λ = 0 (Lagrange-multiplier V) • < 0 large mass limit (quadratic V) H ⇒ in these subspaces, the KS model matches EM Example: KS Bumblebee model in Riemann spacetime

field strength 

quadratic potential 

matter current 

timelike vev 

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

No conventional Higgs mechanism Riemannn spacetime

Theory can have a massive mode how does it affect gravity? Equations of motion:

where

NG modes alone obey Einstein-Maxwell eqs

massive mode obeys

massive mode acts as source of charge & energy

has nonlinear couplings to gravity and Bµ equations can’t be solved analytically 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

To illustrate the behavior of the NG & massive modes, it suffices to work with linearized equations of motion linearized theory is stable in flat-spacetime limit massive mode acts as source of charge & energy equations can be solved

get that static massive mode

the massive mode acts as a static primordial charge density that does not couple with matter current Jµ Fate of NG modes

Find that the diff NG mode drops out of and

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

Find that the Lorentz NG modes propagate

Lorentz NG excitations obey axial gauge condition

removes massive mode from propagating degrees of freedom

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

§ Bjorken (1963) – composite fermion models collective fermion excitations give rise to composite photons emerging as NG modes § Nambu (1968) - local U(1) vector theory in nonlinear gauge has a nonzero vev for the EM field classically equivalent to

Neither gives signals of physical Lorentz violation

Here the KS bumblebee model is different has no local U(1) gauge invariance NG modes behave like photons has signatures of physical Lorentz violation includes gravity (local Lorentz symmetry) 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 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

Special cases: (i) no charge couplings

and decouple from matter

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

e.g., with

Newton’s constant rescales (ii) no massive mode

clearly the most natural choice

and usual electromagnetic fields

usual Newtonian potential

(iii) heavy massive mode

same solutions emerge with a massive mode when large mass limit

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 Higgs Mechanism Riemann-Cartan Spacetime: and

dynamical spin connection

and (tetrad postulate)

To quadratic order, the kinetic term becomes

ab quadratic “mass” terms in ωµ ab Suggests a Higgs mechanism is possible for ωµ Note: Only works in the context of a theory with ab dynamical torsion allowing propagation of ωµ Can get a Higgs mechanism in Riemann-Cartan spacetime Model Building in Riemann-Cartan Spacetime:

ab consider propagating ωµ in a flat background

ab need to add a kinetic term for ωµ  Ghost-free models are extremely limited the massless modes must match with

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

Cardinal Model

symmetric 2-tensor Cµν in Minkowski space with SLV 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

Phon Model

anti-symmetric 2-tensor Bµν coupled to gravity with SLV up to 4 NG modes called phon modes (phonene) certain models produce a scalar (inflaton scenarios) massive modes exist that can modify gravity V. Conclusions In gravity models with spontaneous Lorentz breaking diffeomorphisms also spontaneously broken both NG and massive modes can appear

Gravitational Higgs effect depends on the geometry -Riemann-Cartan spacetime: 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 Bumblebee Models NG modes propagate like massless photons massive mode modifies Newtonian potential Einstein-Maxwell solution is special case Open Issues & Questions Physically viable models with SLV? è must eliminate ghosts è quantization è Higgs mechanism with massive spin connection è models with signatures of 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