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 dEu

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.

• Each mode ϕn breaks the scale invariance.

• In the limit N X lim N→∞ n=0 the scale invariance is recovered.

Unparticles and Cosmology, October 23rd, 2008, Universitaet Wuerzburg 16 The deconstruction for t → uOU decay

∞ λ µ λ X µ i uγ¯ µ(1 − γ5)t ∂ OU −→ i uγ¯ µ(1 − γ5)t Fn∂ ϕn dU dU ΛU ΛU n=0 ⇓ λ2 m E2 m2 − m2 A t u 2 t n 2 dU 2 2 dU −2 Γ(t → uϕn) = Fn with Eu = and Fn = ∆ (mn) 2dU 2π 2m 2π ΛU t 2mt Number of states |ϕni in the interval (Eu,Eu + dEu): dN = dEu ∆2

⇓ dΓ 2m λ2 m2 t t 2 2 (dU −2) = Γ(t → u + ϕn) = Ad Eu(mt − 2mtEu) θ(mt − 2Eu) dE ∆2 2dU U 2π2 u ΛU The same as the Georgi’s result!

Unparticles and Cosmology, October 23rd, 2008, Universitaet Wuerzburg 17 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 ∞ ∞ 1 X X δ V = m2 ϕ2 + κ |H|2 F ϕ 2 n n U n n n=0 n=0 κ v2F A U n 2 2 2 dU 2 2 dU −2 hϕni = − 2 for h|H| i = v ,Fn = ∆ (mn) mn 2π So, ∞ Z ∞ 2 X 2AdU dm hOU i = Fnhϕni −→ −κU v = −∞ 2π (m2)3−dU n=0 0

Unparticles and Cosmology, October 23rd, 2008, Universitaet Wuerzburg 18 • The IR divergence!

0 2 P 2 • A possible regularization δ V = ζ|H| ϕn is not scale invariant. Since the scaling invariance is anyway violated by the vacuum expectation value 6= 0 2 through |H| OU so we adopt

0 2 X 2 δ V = ζ|H| ϕn n as the IR regulator. Then

2 κU v vn = hϕni = − 2 2Fn mn + ζv

The minimization for H reads:

2 2 X X 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 (µU ) v = 0

Unparticles and Cosmology, October 23rd, 2008, Universitaet Wuerzburg 19 dU Ad for λ ≡ ζdU −2Γ(d − 1)Γ(2 − d ) and (µ2 )2−dU ≡ κ2 U U 4 U U U U 2π 2dU −1 2 2 4 2 2−dU 2dU Veﬀ = m |H| + λ|H| − λU (µU ) |H| dU 2 Even if m = 0 one can get the vacuum expectation value 6= 0 (ΛU provides the scale):

1 1 dBZ−2 2 2−dU 2−dU 2−dU 2 λU 2 2 AdU ΛU 2 v = µU for µU = 2 ΛU λ 2π MU

Unparticles and Cosmology, October 23rd, 2008, Universitaet Wuerzburg 20 UnCosmology B. G. and Jose Wudka, “UnCosmology,” arXiv:0809.0977

• The equation of state for unparticles

The trace anomaly of the energy momentum tensor for a gauge theory with massless fermions: β θ µ = N [F µνF ] (1) µ 2g a a µν where β denotes the beta function and N stands for the normal product.

Non-trivial IR ﬁxed point at g = g?, so in the IR we assume

β = δ(g − g?), δ > 0 in which case the running coupling reads

δ δ g(µ) = g? + cµ ; β[g(µ)] = δcµ where c is an integration constant and µ is the renormalization scale.

Unparticles and Cosmology, October 23rd, 2008, Universitaet Wuerzburg 21 From the thermal average of (1) choosing the renormalization scale µ = T and µ using hθµ i = ρU − 3pU , we get

β µν 4+γ ρU − 3pU = hN [Fa Fa µν]i = AT 2g?

4+γ ρU − 3pU = AT ⇓ 3 T 4 A ρ = σT 4 + A 1 + T 4+γ and p = σ + T 4+γ U γ U 3 γ where σ is an integration constant. ⇓ p = 1ρ 1 − Bργ/4 for B ≡ A U 3 U U σ1+γ/4

−γ One can expect that A ∝ ΛU , therefore we obtain γ ( T π2 gIR + f for T < ΛU 4 ΛU ∼ ρNP = 30T × gBZ for T >∼ ΛU

Unparticles and Cosmology, October 23rd, 2008, Universitaet Wuerzburg 22 2 7 where gBZ = 2(nc − 1 + 8ncnf ) for SU(nc) with nf ﬂavours in the BZ sector.

• From the continuity at T = ΛU , the constant f could be determined: f = gBZ −gIR.

• We will assume gBZ ∼ gIR.

Unparticles and Cosmology, October 23rd, 2008, Universitaet Wuerzburg 23 γ ( T π2 gIR + f for T < ΛU 4 ΛU ∼ ρNP = 30T × gBZ for T >∼ ΛU Deconstruction (Stephanov’07):

∞ X 2 2 OU → Fnϕn with mn = ∆ n n=0

The above result ﬁts the following guess for the eﬀective number of degrees of freedom: 2 R T 2 2 2 2 0 dM ρ(M )θ(ΛU − M ) gU (T ) ∝ 2 R ΛU 2 2 0 dM ρ(M ) where ρ(M 2) ∝ (M 2)(dU −2). Then

T 2(dU −1) gU (T ) ∝ ΛU

=⇒ In the presence of just one unparticle operator one can argue that γ = 2(dU − 1).

Unparticles and Cosmology, October 23rd, 2008, Universitaet Wuerzburg 24 • Freeze-out and thaw-in

♣ Brief history of the Universe in the presence of unparticles (no mass-gap).

• T MU : the BZ sector in form of massless particles (no unparticles yet), thermal equilibrium with the SM is maintained (assumption), so T = TBZ = TSM

• 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 extra sector (which at higher temperature, T >∼ ΛU , is made of the BZ matter, while below ΛU of unparticles).

Let’s denote by Tf the decoupling temperature at which

Γ(SM ↔ NP ) ' H where H is the Hubble parameter

2 8π H = 2 ρtot(T ) for ρtot = ρSM + ρNP 3MP l

Unparticles and Cosmology, October 23rd, 2008, Universitaet Wuerzburg 25 There are 2 interesting cases:

• MU > Tf > ΛU :

– Tf is determined by the condition

Γ(SM ↔ BZ) ' H

– For T > Tf the SM and the BZ sectors evolve in thermal equilibrium, but even for T < Tf their temperatures remain equal (T = TBZ = TSM) since ΛU > v.

• ΛU > Tf :

– Till T = ΛU the SM and unparticles still have the same temperature. – For ΛU >∼ T >∼ Tf still the equilibrium is maintained (assumption, in general this depends on dU ). The decoupling temperature Tf must be now determined by

Γ(SM ↔ OU ) ' H

– Till T ∼ v temperatures of SM and unparticles remain equal, at T ∼ v they split. =⇒ The unparticle cosmic background should be there.

Unparticles and Cosmology, October 23rd, 2008, Universitaet Wuerzburg 26 ♣ The Banks-Zaks phase.

1 † LBZ = H H (¯qBZqBZ) MU Then

T 3 T 2 < ΓBZ ∝ 2 and H ∝ =⇒ decoupling for T ∼ Tf−BZ MU MP l ♣ The unparticle phase.

dBZ−dU ΛU LU = k OU OSM for k = dSM + dBZ − 4 MU The most relevant operators for scalar unparticles are

3−dU 3−dU 3−dU ΛU † ΛU ¯ ΛU µν Ls = H H OU , Lf = 3 `He OU , Lv = 3 (BµνB ) OU MU MU MU

3 2dU −3 2 ΛU T T Ls =⇒ ΓU ∝ 2 and H ∝ =⇒ Tf−U MU ΛU MP l

Unparticles and Cosmology, October 23rd, 2008, Universitaet Wuerzburg 27 Γ 5 U 2dU −5 dU > 2 decoupling for T < Tf−U freeze-out ∝ T =⇒ 5 H dU < 2 decoupling for T > Tf−U thaw-in

Figure 1: Regions of (MU , ΛU ) for various Figure 2: Regions of (MU , ΛU ) for various scenarios of decoupling for dU = 3/2. scenarios of decoupling for dU = 3.

decoupling in BZ phase decoupling in the unparticle phase green + + blue − + yellow + − red − −

Unparticles and Cosmology, October 23rd, 2008, Universitaet Wuerzburg 28 • BBN constraints

+0.10 Big-Bang Nucleosynthesis =⇒ ∆Nν = −0.37−0.11 =⇒ upper limit for gIR

† • Assume freeze-out above the EW scale (Tf > v = 246 GeV, L ∝ H HOU )

" #4/3 π2 g(s) (T ) 7 π2 ρ (T ) = g T 4 SM BBN and ρ (T ) = ∆N T 4 U BBN IR 30 BBN (s) U BBN 4 ν 30 BBN gSM (T = v)

⇓ < gIR ∼ 4.3 at 4σ 2 7 To be compared with e.g. gBZ = 2(nc − 1 + 8ncnf ), for nc = 3 and nf = 10, gBZ ' 60.

µν • Assume freeze-out below TBBN (Tf-U < TBBN , L ∝ BµνB OU )

7 g = ∆N =⇒ g < 0.05 at 4σ IR 4 ν IR ∼

Unparticles and Cosmology, October 23rd, 2008, Universitaet Wuerzburg 29 Summary

• Intensive activity on unparticles (∼ 200 citations of the ﬁrst Georgi’s paper)

• Unparticles could be deconstructed

• Troubles with IR divergences

• Cosmological consequences

1 h δ/4i – Rough arguments for the equation of state for unparticles: pU = 3ρU 1 − BρU – Rough arguments for the energy density for unparticles ”derived”:

 δ 2 T π 4  gIR + (gBZ − gIR) for T < ΛU ρ = T × ΛU ∼ NP 30  gBZ for T >∼ ΛU – Unparticles in equilibrium: freeze-out and thaw-in. – BNN bounds on the number of degrees of freedom for unparticles.

• HEIDI

Unparticles and Cosmology, October 23rd, 2008, Universitaet Wuerzburg 30 Experimental constraints From A. Freitas and D. Wyler,“Astro Unparticle Physics”, arXiv:0708.4339.

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 MZ MZ MZ MZ cP1 ¯ cP2 ¯ µ + fD/ γ5f OU + fγµγ5f ∂ OU . dU dU MZ MZ

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 ci.

Unparticles and Cosmology, October 23rd, 2008, Universitaet Wuerzburg 31 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].

Unparticles and Cosmology, October 23rd, 2008, Universitaet Wuerzburg 32 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

1 X i2π~k~y φ(x, y) = √ φ~ (x)e L 2Lδ/2 k ~k

• Extra terms in the scalar potential

λ V (H, φ) = · · · − 1(2f φ − |H|2) 8 1

Unparticles and Cosmology, October 23rd, 2008, Universitaet Wuerzburg 33 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)

Unparticles and Cosmology, October 23rd, 2008, Universitaet Wuerzburg 34