Introduction to Unparticles

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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 representation „ « 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 massless 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. for λ = cU dU M k ΛU U The diﬀerential decay rate |M|2 dΓ = dΦU dΦu 2mt ⇓

2 2 2 dΓ Ad m E |λ| θ(m − 2E ) = U t u t u 2d 2 2−d dEu 2 U (m − 2mtEu) U 2π ΛU t ⇓ „ « „ « dU −2 2 1 dΓ 2 Eu Eu mt = 4dU (dU − 1) 1 − 2 Γ dEu mt mt

16 10 dlnΓ m t d E u

dU =2 6

4 7/3 8/3 2 5/3 3 4/3

0.1 0.2 0.3 0.4 0.5

Eu m t

17 Couplings of unparticles to the SM

Assumptions:

• OU in neutral under the SM gauge group

• dim(OSM) ≤ 4 ΛdBZ −dU L = c U O O for k = d + d − 4 int U k U SM SM BZ MU

• Scalar unparticles OU – Couplings with gauge bosons

−dU µν −dU µν −dU µν λggΛU G Gµν OU , λwwΛU W Wµν OU , λbbΛU B Bµν OU , ˜ −dU ˜µν ˜ −dU ˜ µν ˜ −dU ˜µν λggΛU G Gµν OU , λwwΛU W Wµν OU , λbbΛU B Bµν OU ,

– Coupling with Higgs and Gauge bosons

2−dU † ˜ −dU † µ λhhΛU H HOU , λhhΛU (H DµH)∂ OU , −dU † 2 −dU † µ λ4hΛU (H H) OU , λdhΛU (DµH) (D H)OU ,

– Couplings with fermions and gauge bosons

−dU ¯ µ −dU ¯ µ −dU ¯ µ λQQΛU QLγµD QLOU , λUU ΛU URγµD UROU , λDDΛU DRγµD DROU , −dU ¯ µ −dU ¯ µ −dU µ λLLΛU LLγµD LLOU , λEE ΛU ERγµD EROU , λνν ΛU ν¯RγµD νROU , ˜ −dU ¯ µ ˜ −dU ¯ µ ˜ −dU ¯ µ λQQΛU QLγµQL∂ OU , λUU ΛU URγµUR∂ OU , λDDΛU DRγµDR∂ OU ,

18 ˜ −dU ¯ µ ˜ −dU ¯ µ ˜ −dU µ λLLΛU LLγµLL∂ OU , λEE ΛU ERγµER∂ OU , λRRΛU ν¯RγµνR∂ OU , 1−dU C λYRΛU ν¯R νROU ,

– Couplings with fermions and Higgs boson

−dU ¯ −dU ¯ ˜ λYU ΛU QLHUROU , λYDΛU QLHDROU , −dU ¯ −dU ¯ ˜ λY ν ΛU LLHνROU , λYE ΛU LLHEROU ,

µ • Vector unparticles OU – Couplings with fermions

0 1−dU ¯ µ 0 1−dU ¯ µ 0 1−dU ¯ µ λQQΛU QLγµQLOU , λUU ΛU URγµUROU , λDDΛU DRγµDROU , 0 1−dU ¯ µ 0 1−dU ¯ µ 0 1−dU µ λLLΛU LLγµLLOU , λEE ΛU ERγµEROU , λRRΛU ν¯RγµνROU ,

– Couplings with Higgs boson and Gauge bosons

0 1−dU † µ 0 1−dU µ ν λhhΛU (H DµH)OU , λbOΛU Bµν ∂ O .

s • Spinor unparticles OU

5/2−dU s 3/2−dU ¯s i j λsν ΛU ν¯ROU , λsΛU OU LL²ij H .

19 Unitarity constraints

1. G. Mack, “All Unitary Ray Representations Of The Conformal Group SU(2,2) With Positive Energy,” Commun. Math. Phys. 55, 1 (1977).

2. B. Grinstein, K. Intriligator and I. Z. Rothstein, “Comments on Unparticles,” arXiv:0801.1140 [hep-ph].

⇓

• Im {A(i → i)} > 0 =⇒ dV > 3 for gauge invariant vector unparticle operators • Corrected vector and tensor propagators for unparticles.

20 Deconstruction of unparticles

K¨allen-Lehmanrepresentation of the Feynman propagator: Z Z ∞ dm2 i i∆U (p2) = d4xeipxh0|T {O (x)O (0)}|0i = ρ(m2) F U U 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=1 Then Z X∞ 2 U 2 4 ipx iFn i∆ (p ) = d xe h0|T {OU (x)OU (0)}|0i = F p2 − m2 + iε n=1 n

Ad 2 U 2 2 dU −2 if Fn = 2π ∆ (mn) then

∞ 2 d −2 Z 2 d −2 2 Z 2 Ad X (m ) U Ad (m ) U 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=1 n So, the undeconstructed result has been conﬁrmed. Now, let’s focus on the non-trivial phase: ( ) X∞ F 2 X A n 2 2 2 dU 2 2 dU −2 Im = − Fnπδ(p − mn) −→ − θ(p )(p ) p2 − m2 + iε ∆→0 2 n=1 n n

2 2 So, each peak becomes lower as Fn ∼ ∆ → 0, but their density increases.

21 • Each mode ϕn breaks the scale invariance. • In the limit XN lim N→∞ n=1 the scale invariance is recovered.

The deconstruction for t → uOU decay

X∞ λ µ λ µ i uγ¯ µ(1 − γ5)t ∂ OU −→ i uγ¯ µ(1 − γ5)t Fn∂ ϕn dU dU ΛU ΛU n=1 ⇓

2 2 2 2 λ mtE m − m Ad Γ(t → uϕ ) = u F 2 with E = t n and F 2 = U ∆2(m2 )dU −2 n 2d n u n n U 2π 2mt 2π ΛU Number of states |ϕ i in the interval (E ,E + dE ): dN = dE 2mt n u u u u ∆2 ⇓ dΓ 2m λ2 m2 = t Γ(t → u + ϕ ) = A t E2 (m2 − 2m E )θ(m − 2E ) 2 n 2d dU 2 u t t u t u dEu ∆ U 2π ΛU The same as the Georgi’s result!

22 UnCosmology

(in progress with Jose Wudka) Brief history of the Universe in the presence of unparticles:

• For T À MU the excitations in the new sector are in their BZ phase; they are also in thermal equilibrium with the SM so that T = TBZ = TSM .

• For T <∼ MU : the BZ sector starts to decouple, as the average energy is no longer suﬃcient to create mediators, however, the thermal equilibrium may still be maintained

(T ' TBZ ' TSM ) depending on the strength of eﬀective couplings between the SM and the BZ sector.

• Let Tf be the temperature such that

Γ(Tf ) ' H(Tf )

where Γ is the reaction rate responsible for maintaining the equilibrium between the SM

and the new sectors, while H is the Hubble parameter. For Tf > ΛU Γ is determined by reactions of the form SM↔ BZ between the SM and excitations in the BZ phase. If

Tf < ΛU then Γ is determined by processes of the form SM ↔ U between the SM and the excitations in the U phase.

• BBN constraints

23 Breaking of the scale invariance through OSMOU

(Fox, Rajaraman, Shirman’07)

UV : 1 |H|2 O dBZ −2 BZ MU

⇓

„ d −d « Λ BZ U IR : c U |H|2 O U dBZ −2 U MU 2 2 v ≡ h|H| i= 6 0 ⇒ scaling violation at Λ6 U : „ « dBZ −dU 4−dU ΛU 2−dU 2 Λ6 U = MU v MU

• For Q < Λ6 U unparticles become particles (we assume Λ6 U < ΛU ).

• For Q > Λ6 U unparticle eﬀects could possibly be seen. • Low energy experiments may not be sensitive to unparticles.

24 For the interaction dBZ −dU ΛU cU OSMOU dSM+dBZ −4 MU to cause observable eﬀects, the quantity „ « „ « Λ 2(dBZ −dU ) Q 2(dSM+dU −4) ² ∝ U MU MU must be large enough. However, Q > Λ6 U implies „ « „ « Q 2dSM M 4 ² < U MU v

No ΛU dependence!

• For (g − 2)e, OSM =ee ¯ , Q ' me, so

m6 ² < e < 10−28 for M ≥ 100 GeV 2 4 U MU v

negligible for the present experimental constraint ² < 10−11.

5 • ² ' 1% requires MU <∼ 10 GeV to see unparticle eﬀects at the LHC!

25 Spontaneous symmetry breaking with unparticles and Higgs boson physics

(Delgado, Espinosa, Quiros’07)

UV : 1 |H|2 O dBZ −2 BZ MU ⇓ „ d −d « Λ BZ U IR : c U |H|2 O ≡ κ |H|2 O U dBZ −2 U U U MU

P 2 2 Deconstruction (OU → n Fnϕn, mn = ∆ n) ⇒

2 2 4 Vtot = m |H| + λ|H| + δ V for X∞ X∞ 1 2 2 2 δ V = m ϕ + κU |H| Fnϕn 2 n n n=0 n=0 2 κU v Fn Ad hϕ i = − for h|H|2i = v2,F 2 = U ∆2(m2 )dU −2 n 2 n n mn 2π So, X∞ Z ∞ 2 2 AdU dm hOU i = Fnhϕni −→ −κU v = −∞ 2π (m2)3−dU n=0 0 • The IR divergence!

0 2 P 2 • A possible regularization δ V = ζ|H| ϕn is not scale invariant.

26 Since the scaling invariance is anyway violated by the vacuum expectation value 6= 0 through 2 |H| OU so we adopt X 0 2 2 δ V = ζ|H| ϕn n as the IR regulator. Then κ v2 v = hϕ i = − U F n n 2 2 n 2(mn + ζv ) The minimization for H reads: X X 2 2 2 m + λv + κU Fnvn + ζ vn = 0 n n

Inserting vn one gets in the continuum limit (∆ → 0):

2 2 2 2−dU 2(dU −1) m + λv − λU (µ ) v = 0 d A U dU −2 2 2−dU 2 dU for λU ≡ ζ Γ(dU − 1)Γ(2 − dU ) and (µ ) ≡ κ 4 U U 2π 2dU −1 2 2 2 2−dU 2dU 4 Veﬀ = m |H| + λU (µU ) |H| + λ|H| dU 2 Even if m = 0 one can get the vacuum expectation value 6= 0 (ΛU provides the scale): d −2 „ « 1 „ « 1 ! SM 2−d 2−d 2 2−dU 2 λU U 2 2 AdU U ΛU 2 v = µU for µU = 2 ΛU λ 2π MU

27 2 • Unparticle - Higgs mixing, κU |H| OU , implies Z ∞ (m2)dU −2 iP (p2)−1 = p2 − m2 + v2(µ2 )2−dU r(m2) dm2 h0 U 2 2 2 0 mU (m ) − p + iε

“ 2 ”2 for m2 (m2) = m2 + ζv2 and r(m2) = m U m2+ζv2 250

200

150 mh 100

0 1 1.2 1.4 1.6 1.8 2 dU

The pole Higgs mass mh (lower curve) and the unresummed Higgs mass mh0 (upper curve) 2−d 1/2 as a function of dU for κU = v U and ζ = 1. The straight line is the mgap = ζ v. Figure from Delgado, Espinosa, Quiros’07.

2 • Unparticle - Higgs mixing, κU |H| OU , implies invisible Higgs decays

28 Experimental constraints

From A. Freitas and D. Wyler,“Astro Unparticle Physics”, arXiv:0708.4339 [hep-ph].

CV ¯ µ CA ¯ µ CS1 ¯ CS2 ¯ µ LUff = fγµf O + fγµγ5f O + fD/f OU + fγµf ∂ OU dU −1 U dU −1 U dU dU ΛU ΛU ΛU ΛU CP1 ¯ CP2 ¯ µ + fD/γ5f OU + fγµγ5f ∂ OU dU dU ΛU ΛU (1) cV ¯ µ cA ¯ µ cS1 ¯ cS2 ¯ µ ≡ fγµf O + fγµγ5f O + fD/f OU + fγµf ∂ OU dU −1 U dU −1 U dU dU MZ MZ MZ MZ cP1 ¯ cP2 ¯ µ + fD/γ5f OU + fγµγ5f ∂ OU . dU dU MZ MZ (2)

Here the coeﬃcients have been scaled to a common mass, chosen as the Z-boson mass MZ, so that the only unknown quantities are the dimensionless coupling constants cX. For photons we have

Cγγ µν Cγ˜γ˜ µνρσ LUγγ = − Fµν F OU − ² Fµν Fρσ OU (3) dU dU 4ΛU 4ΛU cγγ µν cγ˜γ˜ µνρσ ≡ − Fµν F OU − ² Fµν Fρσ OU (4) dU dU 4MZ 4MZ

29 Coupling cV cA dU 1 4/3 5/3 2 1 4/3 5/3 2 5th force 7 · 10−24 1.4 · 10−15 1.8 · 10−10 2 · 10−5 4 · 10−24 8 · 10−16 1 · 10−10 1.1 · 10−5 Star cooling 5 · 10−15 2.5 · 10−12 1 · 10−9 3.5 · 10−7 6.3 · 10−15 2 · 10−12 7.3 · 10−10 3 · 10−7 SN 1987A 1 · 10−9 3.5 · 10−8 1 · 10−6 3 · 10−5 2 · 10−11 5.5 · 10−10 1.5 · 10−8 4.1 · 10−7 LEP 0.005 0.045 0.04 0.01 0.1 0.045 0.04 0.008 Tevatron 0.4 0.05 ILC 1.6 · 10−4 1.4 · 10−3 1.3 · 10−3 3.2 · 10−4 3.2 · 10−3 1.4 · 10−3 1.3 · 10−3 2.5 · 10−4 LHC 0.25 0.02 Precision 1 0.2 0.025 1 0.15 0.01 Quarkonia 0.01 0.1 0.45 Positronium 0.25 2 · 10−13 2 · 10−8 0.03

Coupling cS1 cP1, 2cP2 dU 1 4/3 5/3 2 1 4/3 5/3 2 5th force 6.5 · 10−22 1.2 · 10−13 1.6 · 10−8 1.7 · 10−3 — — — — Star cooling 1.3 · 10−9 7 · 10−7 3 · 10−4 0.13 4 · 10−8 1.1 · 10−5 3.3 · 10−3 1 SN 1987A 8 · 10−8 2.4 · 10−6 6.6 · 10−5 2 · 10−3 5.5 · 10−8 1.3 · 10−6 3.5 · 10−5 9 · 10−4 LEP > 1 > 1 > 1 > 1 > 1 > 1 > 1 > 1 ILC > 1 > 1 > 1 > 1 > 1 > 1 > 1 > 1

From A. Freitas and D. Wyler,“Astro Unparticle Physics”, arXiv:0708.4339 [hep-ph].

30 Summary

• Intensive activity on unparticles (170 citations of the ﬁrst Georgi’s paper)

• Interesting and exotic phenomenology

• Unparticles could be deconstructed

• Scale invariance breaking by interactions with the SM (Q > Λ6 U to see unparticles) • Troubles with IR divergences

• Important cosmological consequences

• So far, no fundamental problem of particle physics has been solved by the assumption of unparticles - Is this a drawback?

31 HEIDI

Jochum van der Bij and S. Dilcher:

1. J. J. van der Bij and S. Dilcher, “HEIDI and the unparticle,” Phys. Lett. B 655, 183 (2007) [arXiv:0707.1817 [hep-ph]].

2. J. J. van der Bij and S. Dilcher, “A higher dimensional explanation of the excess of Higgs- like events at CERN LEP,” Phys. Lett. B 638, 234 (2006) [arXiv:hep-ph/0605008].

3. J. J. van der Bij, “The minimal non-minimal standard model,” Phys. Lett. B 636, 56 (2006) [arXiv:hep-ph/0603082].

The model:

• Extra-dimensional (δ) scalars neutral under the SM gauge group X 1 i 2π ~k~y φ(x, y) = √ φ~ (x)e L 2Lδ/2 k ~k

• Extra terms in the scalar potential

λ1 2 V (H, φ) = · · · − (2f1φ − |H| ) 8

32 Similarities:

• The continuous mass spectrum e.g. for s → ∞: ρ(s) ∼ s−3+δ/2

Diﬀerences

• In HEIDI only scalars, while unparticles could have any spin

• Van der Bij and Dilcher don’t assume scale invariance of the extra sector

• In HEIDI interactions between the SM and the extra scalars assumed to be renormalizable

• Van der Bij and Dilcher claim that only for 0 < δ < 1 there is no tachyons in the scalar

spectrum, so the potential is stable (1 < dU < 2)