Unparticle Physics Bohdan GRZADKOWSKI University of Warsaw • Scale invariance and conformal transformations • The scenario – H. Georgi, “Unparticle Physics”, Phys. Rev. Lett. 98, 221601 (2007) • Anomalous breaking of scale invariance and fixed points • Examples of the BZ sector • Correlators and scaling • Phase space for unparticles • Unparticle propagator • Couplings of unparticles to the SM • Deconstruction of unparticles • Spontaneous symmetry breaking with unparticles and Higgs boson physics • UnCosmology – The equation of state for unparticles – Freeze-out and thaw-in – BBN constraints • Summary Unparticles and Cosmology, October 23rd, 2008, Universitaet Wuerzburg 1 Scale invariance and conformal transformations x → x0 = sx Is there a corresponding field transformation such that S = R d4xL is invariant? Assume φ(x) → φ0(x0) = s−dφ(x) where d (the scaling dimension) is to be determined. R 4 1 µ λ 4 4 −2 −2dφ • scalars: Sφ = d x 2∂µφ∂ φ − 4!φ → s s s Sφ ⇒ dφ = 1 | {z1 } R 4 ¯ µ 4 −1 −2dψ 3 • fermions: Sψ = d xψiγ ∂µψ → s s s Sψ ⇒ dψ = 2 | {z1 } • The scaling dimensions and dimensions coincide. • Mass terms are not invariant: – scalars: R d4x m2 φ2 → s4s−2 R d4x m2 φ2 – fermions: R d4x m ψψ¯ → s4s−3 R d4x m ψψ¯ Unparticles and Cosmology, October 23rd, 2008, Universitaet Wuerzburg 2 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 Unparticles and Cosmology, October 23rd, 2008, Universitaet Wuerzburg 3 The scenario UV CFT : BZ fields SM fields ↓ & . 1 MU k OBZOSM 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 Unparticles and Cosmology, October 23rd, 2008, Universitaet Wuerzburg 4 Anomalous breaking of scale invariance and fixed points The scale invariance implies the following Ward identity for 1PI Green’s function Γ(n): ∂ ∂µD = 0 ⇒ − + D Γ(n)(etp , . , etp ) = 0 µ ∂t 1 n−1 (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. Unparticles and Cosmology, October 23rd, 2008, Universitaet Wuerzburg 5 One can use the RGE to verify if the canonical scaling (?) is satisfied: ∂ ∂ ∂ − + β(λ) + (γ − 1)m − nγ(λ) + D Γ(n)(sp ) = 0 ∂t ∂λ m ∂m i for dλ 1 d µ dm β = µ , γ = µ Z , γ = dµ 2 dµ 3 m m dµ In a massless, non-interacting theory the scaling is canonical! 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 t ¯ 0 0 e−n 0 γ[λ(t )]dt = e−nγ[λIR]t and (n) D−nγ(λ ) (n) • Γ (spi) = s IR Γ (pi) Unparticles and Cosmology, October 23rd, 2008, Universitaet Wuerzburg 6 Examples of the BZ sector • Banks & Zaks (1982): SU(3) YM with n massless fermions in e.g. fundamental representation g3 g5 β(g) = − β + β + 3 loops ··· 016π2 1(16π2)2 2 β0 = 11 − 3n 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 g2 33 − 2n β(g ) = 0 for IR = − IR 16π2 306 − 38n Conclusions: – If g = gIR, then the low-energy theory is scale invariant with small anomalous scaling – For n <∼ n0, the theory remains perturbative, so the continuous spectrum doesn’t emerge. Unparticles and Cosmology, October 23rd, 2008, Universitaet Wuerzburg 7 Correlators and scaling Z d4p h0|O (x)O (0)|0i = e−ipxρ (p2) U U (2π)4 U 2 4 R (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 Unparticles and Cosmology, October 23rd, 2008, Universitaet Wuerzburg 8 Phase space for unparticles • In a free scalar quantum field theory Z 3 Z 4 Z d p −ipx d p 0 2 2 −ipx −ipx h0|φ(x)φ(0)|0i = 3 e = 3θ(p )δ(p − m ) e = dΦe (2π) 2Ep (2π) | d{zΦ } • For unparticles Z d4p h0|O (x)O (0)|0i = ρ (p2)e−ipx for ρ (p2) = A θ(p0)θ(p2)(p2)dU −2 U U (2π)4 U U dU ⇓ 4 d pU dΦ (p ) = A θ(p0)θ(p2 )(p2 )dU −2 U U dU U U (2π)4 Note that integrating the phase space for n massless particles one gets Z n ! n 4 X Y d pi (2π)4δ(4) p − θ(p0)δ(p2) = A θ(p0)θ(p2)(p2)n−2 i i (2π)3 n i=1 i=1 for 16π5/2 Γ(n + 1) A = 2 ⇒ A = A n (2π)2n Γ(n − 1)Γ(2n) dU n=dU Unparticles and Cosmology, October 23rd, 2008, Universitaet Wuerzburg 9 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: θ(x) lim = δ(x) for x = p2 and ≡ n − 1 →0+ x1− Unparticles and Cosmology, October 23rd, 2008, Universitaet Wuerzburg 10 Unparticle propagator The Feynman propagator: Z Z ∞ 2 U 2 4 ipx dm 2 i i∆F (p ) = d xe h0|T {OU (x)OU (0)} |0i = ρU (m ) 2 2 0 2π p − m + iε 2 2 2dU −2 From the scaling properties ρU (m ) = AdU θ(m ) m , so U 2 AdU 1 ∆F (p ) = 2 2−d 2 sin(πdU )(−p − iε) U • Non-trivial phase: Ad Im ∆U (p2) = − U θ(p2)(p2)dU −2 F 2 • Interference with the Z-boson: 2 1 ∆Z(p ) = 2 2 p − mZ + iMZΓZ U 2 • Im ∆F (p ) 6= 0 doesn’t imply unparticle decay! Unparticles and Cosmology, October 23rd, 2008, Universitaet Wuerzburg 11 An example: t → u OU dBZ λ µ ΛU Lint = i uγ¯ µ(1 − γ5)t ∂ OU + H.c. for λ = cU dU M ΛU U The differential decay rate |M|2 dΓ = dΦU dΦu 2mt ⇓ dΓ A m2E2|λ|2 θ(m − 2E ) = dU t u t u dE 2 2dU (m2 − 2m E )2−dU u 2π ΛU t t u ⇓ dU −2 2 1 dΓ 2 Eu Eu mt = 4dU (dU − 1) 1 − 2 ΓdEu mt mt Unparticles and Cosmology, October 23rd, 2008, Universitaet Wuerzburg 12 dlnΓ 10 m t d E u 8 d =2 U 6 4 7/3 8/3 2 5/3 3 4/3 0.1 0.2 0.3 0.4 0.5 E m u t Unparticles and Cosmology, October 23rd, 2008, Universitaet Wuerzburg 13 Couplings of unparticles to the SM Assumptions: • OU in neutral under the SM gauge group • dim(OSM) ≤ 4 dBZ−dU k ΛU ΛU 4−dSM−dU Lint = cU k OU OSM ∝ ΛU OU OSM for k = dSM + dBZ − 4 MU MU 2−dU † • Scalar unparticles OU : ∝ ΛU H HOU for dSM = 2 s 5/2−dU s • Spinor unparticles OU : ∝ ΛU ν¯ROU for dSM = 3/2 Unparticles and Cosmology, October 23rd, 2008, Universitaet Wuerzburg 14 Deconstruction of unparticles K¨allen-Lehmanrepresentation of the Feynman propagator: Z Z ∞ 2 U 2 4 ipx dm 2 i i∆F (p ) = d xe h0|T {OU (x)OU (0)}|0i = ρ(m ) 2 2 0 2π p − m + iε 2 2 2 dU −2 with ρU (m ) = AdU θ(m )(m ) . Deconstruction (Stephanov’07): ∞ X 2 2 OU → Fnϕn with mn = ∆ n n=0 Then ∞ Z X iF 2 i∆U (p2) = d4xeipxh0|T {O (x)O (0)}|0i = n F U U p2 − m2 + iε n=0 n Ad 2 U 2 2 dU −2 if Fn = 2π ∆ (mn) then ∞ 2 dU −2 Z 2 dU −2 2 Z 2 Ad X (m ) Ad (m ) dm dm i i U n ∆2 → i U = ρ(m2) 2π p2 − m2 + iε ∆→0 2π p2 − m2 + iε 2π p2 − m2 + iε n=0 n Unparticles and Cosmology, October 23rd, 2008, Universitaet Wuerzburg 15 Let’s focus on the non-trivial phase: ( ∞ ) F 2 A X n X 2 2 2 dU 2 2 dU −2 Im = − Fnπδ(p − mn) → − θ(p )(p ) p2 − m2 + iε ∆→0 2 n=0 n n 2 2 So, each peak becomes lower as Fn ∼ ∆ → 0, but their density increases.