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 special relativity - field theories invariant under global LTs
local ⇒ symmetry of general relativity - Lorentz symmetry holds locally
Previous talk looked at how to construct the SME in the presence of gravity
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 • spacetime 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 spin connection 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 gauge theory
→ 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 & string theory
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
tensor vev
hµ⌥ = h⌥µ
µ⌥ = ⌥µ 1 1 µ Tµ⌥ tµ⌥ kinetic = R Bµ B ⇤ ···⌅⇥ ··· L 16⇥G 4 1 eµ⌥ = ⇧µ⌥ +( hµ⌥ + µ⌥) ⌅T⌃µ⌥ 2 =(T⌃µ⌥ t⌃µ⌥ ) Bµ = DµB D Bµ ··· ··· ··· 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 + a4DµB D B ⌃ µµ⌥⌥ = L⌥µ a b 16 ⇥⇤⇥G 2 T ···gg⌃ µ ⇧g=µ⇥egµ⌥⇤e...T⇧ ⇤ab ··· = t µ µ Bµ+=a5bDµ µ+BA Dµ B + a6DµB D B ⇥ 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 tensor a1R fields+ a2B µTB =0R + a3B B Rµ + a4DµB D 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) ⇥⇤ d µ 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··· ( ⌃ )T µ⇧ ( 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 ⌃µ T⌥⌥ 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 ) ( µ t⌥) + 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 hµ b )=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 Bµ B µ L 16⇥G 4 DµJ =0
µ Bµ = DµB D Bµ 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 DµB D 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 invariant a R + a BµB R + a B B Rµ + a DµB D B T Tabc d a e b 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 µB D B + a6DµB D B ⇥ 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⌅ (⌃⇤⇧ )T µ⌅ (⌃µ⇧d )T⇤ ⌅ (b⌃e=⌅⇧⇧⇥µ)TbT+⇤ µ⇤ µ¯b+(i )kµ⌥ ⇧ (⌃ T⇤µ⌅) ⇥ T abc Tabc + ⇤a Tdbc + ⇤ab MTkaecxa +xa +···⌃ ··· ⇥ ··· ···L ⇧ ···⇥⌃ ··· T⇤µ⌅... - spacetimeT⇤µ⌅... (diffeomorphisms⌃⇤⇧ )Td µ⌅ e... f (⌃ µ ⇧ x)µT⇤ ⌅µ x...⌃µ + ⌃µ µd⌃ ⇧ (⌃ T⇤µe⌅...) f T T ⇥ (⌥ ⌃Tabc) T µ⇧ (a⌥µ ⌃b )Tc Tdef (⌥[⇧x⌃T,xabc)T ]= +···i+⇤⌅a Tdbc ⌃+(⌥⇤b TTaec ) + ⇤c Tabf + ⌅µ⇧ ⇥ ⌅µ⇧ ⌅ ··· ⇥ ⌅ ⇧ ··· ⇤⇥ ···⌅µ ··· ··· ⌅µ⇧··· ··· ··· d e f 1 ⇥ d µ e f TT⇤abcµ⌅... T ⇤aµ ⌅ ...bT c(⌃T⇤⇧def)T µ⌅ T...abc+iq⇤(⌃d⌅+µT⇧⇤aF)TT⇤ ⌅dbc+ ⇤¯⇤...e+T D⇤b T aec+ ⇤ f+T⇤c Tabf+ + T T ⇥⇥(⌥ ⌃ )T abcµ⇧... (⌥abcµ⇤⌃ )T a dbc ⇥ b⌃ aec(⌥···µ T ···c ) abf ··· ⌅µ⇧... ⇥ ⌅µ⇧...··· ⌅ ··· ⇥ ···L ··· 4⌅ ⇧··· ... ······ ··· ··· ⌅µ⇧... ··· ······ d e f leave theTabc lagrangianT abcT 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)T µ⇧+ ⇤a(⌥Tµdbc⌃ )T+⌅ ⇧⇤b Taec(⌥⇧⌃ +)T⌅µ + ⌃ (⌥ T⌅µ⇧) When is local⇥ Lorentz ···symmetry⇥ ··· spontaneously ··· broken? ··· ··· ··· T⌅µ⇧ T⌅µ⇧ (⌥⌅T⌃ )T µ⇧ (⌥ (⌃⌥µ)⌃T )µT⇧⌅ ⇧... (⌥(⌥µ⇧⌃⌃ )T)T⌅µ + ⌃⌃ ((⌥⌥ TT⌅µ⇧) ) ⇥ ⌅µ⇧... ⇥ ⌅µ⇧... ⌅ ⌅ ⇧... ······ ⌅µ⇧... T⌅µ⇧... T⌅µT⇧... (⌥⌅⌃T)T µ⇧... (⌥ (⌃⌥µ)⌃T )µT⇧⌅ ⇧...... (⌥µ⌃ )T ⌃ (⌥ T⌅µ⇧...) ⇥ ⌅µ⇧ ... ⇥ ⌅µ⇧... ⌅ ··· ⌅ ⇧... ··· T T (⌥ ⌃ )T µ⇧... (⌥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) 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
(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 (photons, gravitons) 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
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 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
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 ⌅ Bµ B + ⌅ DµB D B + ⌅ DµB D B 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 ⌅ Bµ B + ⌅ DµB D B + ⌅ DµB D B 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 ⌅ Bµ B + ⌅ DµB D B + ⌅ DµB D B 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 in B a vector-tensor theory of gravity L L4int 1 µ B = setB µ B 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 =0Bµ BL ⇤⇧ LB L 4 R = 0 10V +35 mint int L=0PlanckL ⇤ L L int⇧ L = 35⇥0V ⌅V=0+ RPlanck L10 L0 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 ⌅ Bµ B + ⌅ DµB D B + ⌅ DµB D B 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 ⌅1Bµ B + ⌅2DµB D B + ⌅3DµB D B L models with4 Will-Nordvedt2 kinetic terms2 µ µ =+⇤ B B Rµ + ⇤ B BµR LB 1 2 1 µ 1 µ 1 µ ⌅ Bµ B + ⌅ DµB D B + ⌅ DµB D B 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 Bµ B Kostelecky-SamuelLB 16⇥G µ4 models 1 µ no propagating = Bµ B LB 4 ghost modes
charged matter interactions Lint with =0global 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 electromagnetism
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 photon 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