Aspects of Spontaneous Lorentz Violation

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 local symmetry of special relativity - field theories invariant under global LTs 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 T λµν... spacetime tensor components To reveal the local Lorentz symmetry, introduce local tensor components in Lorentz frames T abc... local Lorentz frame components

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

Can also introduce torsion T λ µν = Γ λ µν - Γ λ νµ spin connection becomes dynamically independent gives gravity the form of a gauge theory 16 components 24 components New geometry emerges: Riemann-Cartan spacetime curvature = R κ λµν torsion = T λ µν 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 But with spontaneous LV No-go theorem: act as fixed background fields in any observer frame arise as vev s must be treated dynamically 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., a µ, b µ, c µν, d µν, H µν,... 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 f the physical vacuum picks one, breaking the 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 const. scalar field (electroweak) V <f> 0 f tensor vev <T> 0 vacuum breaks Lorentz symmetry

How is SLV introduced? Consider a Lorentz-invariant lagrangian L with tensor fields T include a potential that occurs when T V that has a nontrivial minimum has a nonzero vev T =0 e.g., in flat spacetime, with components T abc V = V (T abc ad be cf...t def t 2 ) has a minimum when T abc t abc = 0 where t 2 = t abc ad be cf...t def What about in curved spacetime? Lorentz symmetry is a local symmetry

Use a vierbein description in curved spacetime vierbein connects spacetime tensors to tensors in local Lorentz frame spacetime components e.g., local frame components 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)

In curved spacetime, the Lagrangian is invariant under both local Lorentz transfs and diffeomorphisms b - rotations & boosts in local frame a = a b T abc T abc + d a T dbc + e b T aec + - spacetime diffeomorphisms x µ x µ + µ + b T µ... T µ... ( )T µ... ( µ )T... L a leave the lagrangian invariant L L When is local Lorentz symmetry spontaneously broken?

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 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 (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 SLV to address these questions: Bumblebee models Cardinal models B µ A µ photons? C µ g µ gravitons? Antisymmetric two-tensor models

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 Vacuum 10 symmetric comps. 6 antisymmetric comps. 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> = 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 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: 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 suggests a Higgs mechanism is possible for ω µ ab only works with dynamical torsion allowing propagation of ω µ ab 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 A µ 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

Bumblebee Lagrangian: L = 1 16 G (R 2 )+L B V (B µ B µ ± b 2 )+L int minimum of V gives the vev Have different choices for the kinetic, potential, & int terms depending on the interpretation of the vector For B µ L vector in a vector-tensor theory of gravity set L int =0 gravitational couplings only Or for B µ L generalized vector potential (photons?) keep L int =0 allows Lorentz violating matter ints

Bumblebee Kinetic Terms: (1) B µ as in a vector-tensor theory of gravity models with Will-Nordvedt kinetic terms L B =+ 1 B µ B R µ + 2 B µ B µ R 1 4 1B µ B µ + 1 2 2D µ B D µ B + 1 2 3D µ B µ D B expect propagating ghost modes (2) B µ as a generalized vector potential Kostelecky-Samuel models L B = 1 4 B µ B µ no propagating ghost modes charged matter interactions with global U(1) charge

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

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 H H < 0 exist ultimately means bumblebee models are useful at low energy as effective or approx theories KS models can perform a Hamiltonian constraint analysis can find subspace of phase space with λ = 0 (Lagrange-multiplier V) large mass limit (quadratic V) H b µ =(b, 0, 0, 0) H > 0 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 B µ 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 J µ

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 Special cases: (i) no charge couplings 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 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 (iii) heavy massive mode usual Newtonian potential 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 b µ signal physical Lorentz breaking

Higgs Mechanism Riemann-Cartan Spacetime: and dynamical spin connection and (tetrad postulate) To quadratic order, the kinetic term becomes quadratic mass terms in ω µ ab Suggests a Higgs mechanism is possible for ω µ ab Note: Only works in the context of a theory with dynamical torsion allowing propagation of ω µ ab Can get a Higgs mechanism in Riemann-Cartan spacetime

Model Building in Riemann-Cartan Spacetime: consider propagating ω µ ab in a flat background need to add a kinetic term for ω µ ab Ghost-free models are extremely limited the massless modes must match with Results for ghost-free models: models with propagating massless ω µ 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 Cardinal Model symmetric 2-tensor C µν 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 B µν 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