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Introduction to Unparticles Nordic Workshop on LHC and Beyond, 12-14 June 2008, Stockholm Introduction to Unparticles Bohdan GRZADKOWSKI Institute of Theoretical Physics, University of Warsaw H. Georgi, PRL 98, 221601 (2007), cited 170 times: SM ⊗ CFT(BZ) | {z } ⇓ OSM OBZ k MU ⇓ Dimensional transmutation in the BZ sector (breaking of the scale invariance at µ = ΛU ) ⇓ IR fixed point (the scale invariance emerges at the loop level) 1 1. Scale invariance and conformal transformations 2. Anomalous breaking of scale invariance and fixed points 3. Mass spectrum in scale invariant theories 4. Dimensional transmutation 5. Examples of the BZ sector 6. Correlators and scaling 7. Phase space for unparticles 8. Unparticle propagator 9. Couplings of unparticles to the SM 10. An example: t → u OU 11. Unitarity constraints 12. Deconstruction of unparticles 13. Breaking of the scale invariance through OSMOU 14. Spontaneous symmetry breaking with unparticles and Higgs boson physics 15. Unparticles and AdS/CFT 16. UnCosmology 17. Experimental Constraints 2 Scale invariance and conformal transformations x → x0 = sx R Is there a corresponding field transformation such that S = d4xL is invariant? Assume φ(x) → φ0(x0) = s−dφ(x) where d (the scaling dimension) is to be determined. • scalars: Z ff 4 1 µ λ 4 4 −2 −2d S = d x ∂µφ∂ φ − φ → s s s φ S ⇒ d = 1 φ 2 4! | {z } φ φ 1 • fermions: Z 4 µ 4 −1 −2d 3 S = d xψiγ¯ ∂µψ → s s s ψ S ⇒ d = ψ | {z } ψ ψ 2 1 • The scaling dimensions and dimensions coincide. • Mass terms are not invariant: R R – scalars: d4x m2 φ2 → s4s−2 d4x m2 φ2 R R – fermions: d4x m ψψ¯ → s4s−3 d4x m ψψ¯ 3 R 4 µ The Noether theorem: if S = d xL scale invariant ⇒ ∂ Dµ = 0 for a scalar theory ∂L ν Dµ = (d + xν ∂ )φ − xµL ∂(∂µφ) φ In general (Callan, Coleman, Jackiw (1970)): µ µ ∂ Dµ = Tµ where ∂L 1 2 Tµν = ∂ν φ − gµν L − (∂µ∂ν − gµν )φ ∂(∂µφ) 6 | {z } the canonical energy-momentum tensor µ Scale invariance ⇒ Tµ = 0 1 µ 1 2 2 λ 4 For L = 2 ∂µφ∂ φ − 2 m φ − 4! φ µ 2 2 Tµ = m φ 4 Conformal transformations (angle-preserving) are such that α dx dxα |dx| |dx| remains unchanged, so xµ − bµx2 xµ → x0µ = 1 − 2b · x + b2x2 • conformal invariance ⇒ scale invariance • scale invariance ⇒ conformal invariance for all renormalizable field theories of spin ≤ 1 5 Anomalous breaking of scale invariance and fixed points The scale invariance implies the following Ward identity for 1PI Green’s function Γ(n): „ « µ ∂ (n) t t ∂ Dµ = 0 ⇒ − + D Γ (e p1, . , e pn−1) = 0 ∂t (n) where D = 4 − ndφ is the canonical dimension of Γ . The solution reads (n) D (n) Γ (spi) = s Γ (pi)(?) for s = et. However, since the loop expansion requires some sort of regularization, (so some scale must be introduced: µ in the dimensional regularization, or Λ in the cutoff regularization), therefore one can expect that classical scale invariance (i.e. invariance of the Lagrangian) would be broken at the quantum (loops) level. One can use the RGE to verify if the canonical scaling (?) is satisfied: » – ∂ ∂ ∂ (n) − + β(λ) + (γm − 1)m − nγ(λ) + D Γ (spi) = 0 ∂t ∂λ ∂m for dλ 1 d µ dm β = µ , γ = µ Z3, γm = dµ 2 dµ m dµ In a massless, non-interacting theory the scaling is canonical! 6 In general, there is no scaling in a quantum field theory, even if m = 0. The solution of the RGE reads: R (n) t t D (n) −n t γ[λ¯(t0)]dt0 Γ (e pi, λ, µ) = (e ) Γ (pi, λ¯(t), µ) e 0 • Assume there is an IR fixed point at λ = λIR: so β(λR) = 0. Then • R −n t γ[λ¯(t0)]dt0 −nγ[λ ]t e 0 = e IR and (n) D−nγ(λ ) (n) • Γ (spi) = s IR Γ (pi) The Green’s functions scale with non-canonical scaling dimension d = D − nγ(λIR) µ To check the scale invariance one should find Tµ , e.g. for QCD: β(g) T µ = Ga Ga µν µ 2g3 µν So, if β(g) = 0 then the theory is scale invariant. 7 Mass spectrum in scale invariant theories Operator of the scale transformation: U(²) = ei²D 2 2 i²D 2 −i²D 2² 2 i[D, Pµ] = Pµ ⇒ [D, P ] = −2iP ⇒ e P e = e P 2 2 • Let |pi is a state of momentum pµ: P |pi = p |pi • Assume that the scale invariance is not broken spontaneously: D|0i = 0, then • e−i²D|pi is a state of rescaled momenta: e−i²D|pi ∝ |e²pi. Conclusion: varying ², one can construct states of an arbitrary non-negative mass2: e2²p2. ⇓ Mass spectrum is either continuous or particles are massless 8 Examples of the BZ sector • Banks & Zaks (1982): SU(3) YM with n massless fermions in e.g. fundamental repre- sentation „ « g3 g5 β(g) = − β0 + β1 + 3 loops ··· 16π2 (16π2)2 2 β0 = 11 − 3 n β0(n0) = 0 n0 = 16.5 38 β1 = 102 − 3 n β1(n1) = 0 n1 ' 8.05 If n1 < n < n0 (so β0 > 0 & β1 < 0) then keeping β0 and β1 one gets 2 gIR 33 − 2n β(gIR) = 0 for = − 16π2 306 − 38n ` ε ´ Expanding n around n0: n = n0 1 − 11 g2 IR ' ε × 10−2 16π2 Conclusions: – If g = gIR, then the low-energy theory is scale invariant with small anomalous scaling – For ε ¿ 1, the theory remains perturbative, so the continuous spectrum doesn’t emerge. 9 • Massless Abelian Thirring model (D = 2) µ 2 L = ψiγ¯ ∂µψ + λ(ψγ¯ µψ) – G(x − y) = ih0|T {ψ(x)ψ¯(y)}|0i - exactly calculable – Scaling: G(x − y) = s2dG[s(x − y)] for 1 λ2 1 d = + 2 4π2 λ2 1 − 4π • ”Electro-Magnetic” duality (Intriligator-Seiberg): N = 1 SUSY YM SU(Nc), Nf mass- less quarks in the fundamental representation: 3 – If 2 Nc < Nf < 3Nc then there is a non-trivial IR fixed point and the dual theory is SU(Nf − Nc), N = 1, YM with with Nf quarks. 10 The scenario again UVCFT : BZ fields SM fields ↓ & . 1 MU k OBZ OSM MU ↓ ΛU transmutation scale ↓ ↓ dBZ −dU ΛU IR fixed point cU k OU OSM MU where k = dSM + dBZ − 4. dSM and dBZ are canonical dimensions of OSM and OBZ , respectively, while dU is the scaling dimension (the same as the mass dimension in this case) of OU : 0 0 −dU 0 OU (x) → OU (x ) = s OU (x) with 1 < dU < 2 for x → x = sx An example of matching between OBZ and OU : • (¯qq) in QCD ⇐⇒ M ∝ (¯qq) mesons in the chiral non-linear model 11 Correlators and scaling Z 4 d p −ipx 2 h0|OU (x)OU (0)|0i = e ρU (p ) (2π)2 R 2 4 (4) 2 for ρU (p ) = (2π) dλ δ (p − pλ) |h0|OU (0)|λi| . • Scaling: 0 0 −dU 0 OU (x) → OU (x ) = s OU (x) with 1 < dU < 2 for x → x = sx • Z 2 4 ipx 2 0 2 2 α ρU (p ) = d x e h0|OU (x)OU (0)|0i ⇒ ρU (p ) = AdU θ(p )θ(p )(p ) where AdU is a normalization constant and α is to be determined. ⇓ α = dU − 2 12 Phase space for unparticles • In a free scalar quantum field theory Z Z Z d3p d4p h0|φ(x)φ(0)|0i = e−ipx = θ(p0)δ(p2 − m2) e−ipx = dΦe−ipx 3 3 (2π) 2Ep (2π) | {z } dΦ • For unparticles Z d4p 2 −ipx 2 0 2 2 dU −2 h0|OU (x)OU (0)|0i = ρU (p )e for ρU (p ) = A θ(p )θ(p )(p ) (2π)4 dU ⇓ d4p 0 2 2 dU −2 U dΦU (pU ) = A θ(p )θ(p )(p ) dU U U (2π)4 Note that integrating the phase space for n massless particles one gets ! Z Xn Yn 4 4 (4) 0 2 d pi 0 2 2 n−2 (2π) δ p − θ(p )δ(p ) = Anθ(p )θ(p )(p ) i i (2π)3 i=1 i=1 for 5/2 1 16π Γ(n + 2 ) An = ⇒ A = A (2π)2n Γ(n − 1)Γ(2n) dU n=dU 13 The integrating the phase space for n massless particles one gets ! Z Xn Yn 4 4 (4) 0 2 d pi 0 2 2 n−2 (2π) δ p − θ(p )δ(p ) = Anθ(p )θ(p )(p ) i i (2π)3 i=1 i=1 for 5/2 1 16π Γ(n + 2 ) An = ⇒ A = A (2π)2n Γ(n − 1)Γ(2n) dU n=dU Georgi: Unparticle stuff with scale dimension dU looks like a non-integer number dU of massless particles. The limit n → 1 reproduces the single particle phase space: ²θ(x) lim = δ(x) for x = p2 and ² ≡ n − 1 ²→0+ x1−² 14 Unparticle propagator The Feynman propagator: Z Z ∞ dm2 i i∆U (p2) = d4xeipxh0|T {O (x)O (0)} |0i = ρ (m2) F U U U 2 2 0 2π p − m + iε ` ´ 2 2 2 dU −2 From the scaling properties ρU (m ) = AdU θ(m ) m , so Ad 1 ∆U (p2) = U F 2 2−d 2 sin(πdU ) (−p − iε) U • Non-trivial phase: n o Ad Im ∆U (p2) = − U θ(p2)(p2)dU −2 F 2 • Interference with the Z-boson: 1 ∆ (p2) = Z 2 2 p − mZ + iMZ ΓZ ˘ ¯ U 2 • Im ∆F (p ) 6= 0 doesn’t imply unparticle decay! 15 An example: t ! u OU BZ λ µ ΛU Lint = i uγ¯ µ(1 − γ5)t ∂ OU + H.c.
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