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Aspects of Spontaneous Lorentz Violation

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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 ⇒ 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 <f> ≠ 0 tensor vev <T> ≠ 0 f vacuum breaks Lorentz symmetry 1 eµ⌥ = ⇧µ⌥ +(2hµ⌥ + µ⌥) hµ⌥ = h⌥µ µ⌥ = ⌥µ − 1 1 µν Tµ⌥ tµ⌥ kinetic = R BµνB ⇤ ···⌅⇥ ··· L 16⇥G − 4 1 eµ⌥ = ⇧µ⌥ +( hµ⌥ + µ⌥) ⌅T⌃µ⌥ 2 =(T⌃µ⌥ t⌃µ⌥ ) Bµν = DµBν DνBµ ··· ··· − ··· − Will-Nordvedt hµ⌥ = h⌥µa b c T⌅τµ⌃⇧µ⌥ ==(e⌅ Te⌃µµ⌥e⇧ t⌃Tµabc⌥ ) ······ ··· −···1 ······ µ µ ν µ ν kinetic = a1R + a2BµB R + a3B B Rµν + a4DµBνD B ⌃ µµ⌥⌥ = L⌥µ a b 16⇥⇤⇥G 2 T ···gg⌃µ−⇧g=µ⇥egµ⌥⇤e...T⇧ ⇤ab ··· = t ν µ µ ν Bµ+=a5bDµ µ+BAνDµ B + a6DµB DνB ⇥ How is SLV introduced? 2Tµ⌥ ⌃µ⌥a tµ⌥ a a1 ⇥⇤ ⇤t = ···t ⌅⇥Λ ···b ⇧⌃···δ⇧µb ⇥+⇧⌥⇤⇥ b...t ··· µ µ ν µ ν Consider a Lorentz-invariant lagrangian =with tensora1R fields+ a2B µTB =0R + a3B B Rµν + a4DµBνD B Bµ = bµ + Aµ ⇤ LBµ =16⇥bµG+ Aµ ⌃ ⌥⇧ ⌅T⌃µ⌥ =(⌃Tµ⌃⌥µ⌥ t⌃µ⌥ ) ⇥⇤ν µ 2 µ ν ν 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 ν 6 Tµ Γ ⌦¯(iν↵) ⇥ µ M that occurs when T has=0 a nonzero vev− T =0 −Mk − ± L ⌃ ⌥⇧ ⌃ ⌥⇧ L ⇥ ⌃ ⌥ ⌥µντ⌃µ⌥ =(T⌃µ⌥µ α⇥⇤t⌃µµ⇥⌥ ⇤)µ2 e.g., Vin =flatV (spacetime,T ···g⌥···g Tµwith⌃⇥µg⌥⇤ componentsx...T=···e⌃− xeµ···+e ···⌥⌃Tt...t) ⇥⇤ Λ dΛµ eΛ⌥ f T µ⌥ ⌃ T Γ ⌦¯(i↵)···k ⌅ ⌃ −TabcΓ ⌦¯(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 ⇥ 4α α α α t = t α···⇧⇥ T⇧⇤ ⇧T⌥⇤ ...tt T =0··· (∂ ⌃ )Tαµ⇧ (∂µ⌃ )T (∂⇧⌃ )T ⌃ (∂αT ) has a minimumT = whene e ⌃e µ...t⇥··· ⌅µ⇧ ⇥···⌅µ⇧− ⌅ − ⌅⇧− ⌅µα− ⌅µ⇧ 1⌥µν ⇥ ⌥ µ µ⌃ ν ⌥⇥⇥⇤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= δµ=···aµ⌥⇥ ⌥ + bµ⌥⇥5⇥ ⌥ + DµTν = ⌦µTν Γ⌃µνT⌥⌥ 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,... (DµTν) (Γµνt⌥) + − 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 Tν = e Ta ± α α b ν b b Tµ Tµb (↵µξ )Tα ξ (↵αTµ) Ta ⇥ T Ta + ⌅ TTa ⇥ T Ta + ⌅ T ⇤ a b ⌅ a b ⇤ a b ⌅ ⇤ a −b − (D T )2 ( b ⌅a t )2 + µ ν α µaαν b α α Tµ Tµ (↵µξ ⇥)Tα ξTµ(↵αTTµµ) ···(↵µξ )Tα ξ (↵αTµ) ⇤ − − ⇤ − − µν 2 V = ⌃(Bµg Bν b )=0 ⇤ ± µν 2 V = ⌃(Bµg Bν b ) =0 ⇤ ± ⌅ µ 1 ν b ( µ hµνb )=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 µν kinetic = R BµνB µ L 16⇥G − 4 DµJ =0 µ Bµν = DµBν DνBµ int = BµJ − Will-Nordvedt L 1 19 µ µ ν µ ν b bMPlanckb = hc/G¯ 10 GeV = a R + a BµB R + a B B Rµν + a DµBνD B Λa = ⇥a + ⇤a Lkinetic 16⇤⇥G 1 2 3 4 ν µ µ ν ⌅hc¯ ⇤ 19 +a D B D B + a D B D B µ µ µ ⇤ 5 µ ν 6 µ ν ⇥ M = ⇤ 10 GeV x x + ⌃Planck ⇤ ⇤ ⇥ ⇤ G ⇤ ⇥ 1 d e f d e f µ µ ν µ ν In curved spacetime, the Lagrangian = is invarianta R + a BµB R + a B B Rµν + a DµBνD B T Tabc Λ dΛΛa eΛΛb Λf 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 ··· ··· ··· ⇥ ··· ⇤ ··· ··· ··· ν µ ··· ···µ ν ν 2 d e +a5Df µBνD B + a6DµB DνB ⇥ 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 ···+ΛaT⇤c ==0T⇥abfa +···+⇤a ··· ··· ⇥ ··· ··· ··· ⇧ ⌃⌅ ··· ··· α α α α T⇤µ⌅ T⇤µ⌅ (⌃⇤⇧ )Tαµ⌅ (⌃µ⇧d )T⇤⌅ Λ (b⌃e=⌅⇧⇧⇥µ)TbT+⇤Γµ⇤αψµ¯b+(i )kµ⌥ ⇧ (⌃αT⇤µ⌅) ⇥ T−abc Tabc −+ ⇤a Tdbc +−⇤ab MTkaecxa +xa +···⌃ − ··· ⇥ ··· ···L ∼ ⇧ ···⇥⌃ ··· α α α T⇤µ⌅... - spacetimeT⇤µ⌅...α (diffeomorphisms⌃⇤⇧ )Tdαµ⌅αe... f (⌃ µ ⇧ x)µT⇤⌅µα x...⌃µ + ⌃µ µd⌃ ⇧ (⌃ααT⇤µe⌅...) f T T ⇥ (⌥ ⌃Tabc)−Tαµ⇧ Λ(a⌥µΛ⌃b Λ)Tc−Tdef (⌥[⇧x⌃T,xabc)T ]=−+···i+⇤⌅a−Tdbc ⌃+(⌥⇤bαTTaec ) + ⇤c Tabf + ⌅µ⇧ ⇥ ⌅µ⇧ − ⌅ ··· ⇥− ⌅⇧ −··· ⇤⇥ ···⌅µα ···−··· ⌅µ⇧··· ··· ··· d e f α 1 ⇥αd µ e f TT⇤abcµ⌅... TΛ⇤aµα⌅Λ...bTΛc(⌃T⇤⇧def)Tαµ⌅αT...abc+iq⇤(⌃d⌅+µT⇧⇤aF)TT⇤⌅dbc+⇤¯⇤...e+TαD⇤b Tψaec+ ⇤ f+T⇤c Tabf+ + T T ⇥⇥(⌥ ⌃ )T−αabcµ⇧... (⌥abcµ⇤⌃ )T− a dbc ⇥ b⌃−aec(⌥···µαT ···c ) abf ··· ⌅µ⇧... ⇥ ⌅µ⇧...···− ⌅ ··· ⇥− ···L ∼··· 4⌅⇧··· ..
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