Unparticle physics with broken scale invariance

If scale invariance is exact, unparticles are unlikely to be probed in colliders since there are stringent constraints from astrophysics and cosmology. However these constraints are inapplicable if scale invariance is broken at a scale mu>~ 1 GeV. The case 1 GeV<~ mu<M_Z is particularly interesting since it allows unparticles to be probed at and below the Z pole. We show that mu can naturally be in this range if only vector unparticles exist, and briefly remark on implications for Higgs phenomenology. We then obtain constraints on unparticle parameters from e+ e- ->mu+ mu- cross-section and forward-backward asymmetry data, and compare with the constraints from mono-photon production and the Z hadronic width.


Introduction
Unparticle physics was introduced in Ref. [1] as a low energy effective description of a hidden sector with a nontrivial infrared fixed point. This sector is assumed to interact with the Standard Model (SM) through the exchange of particles at a high scale M . Below M , the interactions are of the form where C i are dimensionless constants, O i SM is an operator with mass dimension d i SM built out of SM fields and O U V is an operator with mass dimension d U V built out of the hidden sector fields. Scale invariance in the hidden sector emerges at an energy scale Λ < M . In the effective theory below Λ the interactions of Eq. (1.1) take the form where d is the scaling dimension of the unparticle operator O.
Unparticle effects might be detectable in missing energy distributions and interference with SM amplitudes [1,2,3,4]. However, if scale invariance is exact, unparticles are unlikely to be probed in colliders since there are strong constraints from astrophysics and cosmology [5,6] (see Section 2). As discussed in Section 3, these constraints are inapplicable if scale invariance is broken at a scale µ 1 GeV, while constraints from experiments at center-ofmass energy √ s > µ remain relevant and resonance-like behavior at µ is expected. Ref. [7] has considered collider phenomenology for µ > M Z . Here we consider the constraints on unparticle parameters assuming 1 GeV µ < M Z which allows unparticles to be probed by s channel Z exchange observables. For scales of Λ and M that are experimentally accessible, the Higgs coupling to scalar unparticles generally breaks scale invariance at the electroweak scale [8,9]. Having µ < M Z in this case requires somewhat small dimensionless couplings (Section 3.1). However, if only vector unparticles exist, scale invariance is broken by higher dimensional operators, and µ can naturally be below M Z (Section 3.2). We also briefly discuss how vector unparticles could affect Higgs phenomenology in Section 3.3.
Constraints on vector and axial-vector unparticle couplings obtained using e + e − → µ + µ − forward-backward asymmetry (FBA) and total cross-section data are presented in Section 4. The resonance-like behaviour at µ is taken into account in the analysis. Our summary is followed by four appendices covering details of the unparticle contribution to the Z hadronic width, the bound from SN 1987A cooling, the vacuum polarization correction to the unparticle propagator, and the initial state QED corrections.

Bounds on vector unparticle interactions
Consider vector unparticles coupling to fermions: which, following the convention of Refs. [2,10], can be written as Using the spectral density ρ( Here (−q 2 ) d−2 is defined as |q 2 | d−2 for negative q 2 and |q 2 | d−2 e −idπ for positive q 2 . A d is chosen following the convention of Ref. [1]: . (2.5) The constant a = 1 if O µ is assumed to be transverse, and a = 2(d− 2)/(d− 1) in conformal field theories [11]. The value of a does not affect the results of this paper. It should be noted that operators of a conformal field theory are subject to lower bounds on their scaling dimensions from unitarity, and in particular d ≥ 3 for vector operators [11,12]. However, this bound can be violated for a hypothetical scale invariant field theory that is not conformally invariant (see e.g. Ref. [13]). We focus on the range 1 < d < 2 since unparticle effects are relatively suppressed for higher values of d. (Also, SM contact interactions induced by messenger exchange at the scale M generally dominate over unparticle interference effects for d ≥ 3 [11].) A bound on the scale of O µ interactions can be obtained from mono-photon production (e + e − → γ + unparticle) at LEP2. The cross-section is given by [3] where c ≡ c 2 V + c 2 A , E γ is the photon energy, and θ γ is the polar angle. Following Ref. [4], we obtain an upper bound on c using the L3 95% C.L. upper limit σ ≃ 0.2 pb (obtained under the cuts E γ > 5 GeV and | cos θ γ | < 0.97 at √ s = 207 GeV) [14]. This "mono-photon bound" corresponds to c < 0.026, 0.032 and 0.057 for d = 1.1, 1.5 and 1.9 respectively. Note that since Λ < M and unparticle effects can only be probed if √ s < Λ, This implies that the current bound from mono-photon production is only relevant for d 2.6 (see Fig. 1). 1 Another process considered in Ref. [3] is Z → qq + unparticle, which contributes to the Z hadronic width. Here we note that it is important to consider the vertex correction together with the real emission process, since the two contributions largely cancel each other for values of d close to 1 and the former contribution dominates for values of d close to 2. As explained in Appendix A, the constraint on unparticles from the Z hadronic width is also weaker than the mono-photon bound. We now compare the mono-photon bound with the constraints on vector unparticles from cosmology and astrophysics [5]. To preserve the successful predictions of Big Bang Nucleosynthesis (BBN), we require the unparticle sector to be colder than SM radiation during BBN, so that its energy density is subdominant. For the operator in Eq. (2.2), the interaction rate Γ ψ redshifts more slowly than the Hubble parameter H if d ≤ 3/2. The unparticle sector can then remain cold if it is decoupled throughout BBN, corresponding to Γ ψ H for T ∼ 1 MeV. For d > 3/2, Γ ψ redshifts faster than H. In this case we require the unparticle sector to decouple before T ∼ 1 GeV so that the QCD phase-transition only heats up SM radiation [5]. The BBN constraint, corresponding to is much more stringent than the mono-photon bound (see Fig. 1). The SN 1987A constraint on unparticle emission [5,6,17,18], where the supernova core temperature is taken to be T SN = 30 MeV and C d ≃ 0.01 (see Appendix B), is similar in magnitude to the BBN constraint.

Broken scale invariance
The BBN and SN 1987A constraints can be evaded provided scale invariance is broken at a scale µ sufficiently large compared to the relevant energy scales (≃ 1 MeV and ≃ T SN respectively). 2 We can model broken scale invariance by removing modes with energy less than µ in the spectral density, so that [8] ρ Due to Boltzmann suppression of the emission, the SN 1987A constraint with scale invariance broken at a scale µ corresponds to replacing C d in Eq. (2.8) by (see Appendix B): Assuming c is close to the mono-photon bound, the SN 1987A constraint can be evaded provided µ 1 GeV. Note that other constraints arising from long range forces [20], contributions to the muon and electron anomalous magnetic moments [3,21], modifications to positronium decay [21], neutrino decay into unparticles [22], and contributions to low energy neutrino-electron scattering amplitudes [23] are also evaded in this case.
The mono-photon bound is also modified when scale invariance is broken: In Eq. (2.6), the numerator inside the parentheses is replaced by s − 2 √ sE γ − µ 2 , and the end-point for E γ is shifted from √ s/2 to (s − µ 2 )/2 √ s. However, these modifications do not change the cross-section appreciably for µ < M Z . Whether scale invariance is broken or not is relevant for the allowed range of the vector unparticle scale dimension d. Consider the decay width from the interaction Eq. (2.2) where an initial fermion with mass m f decays into a massless fermion and the unparticle. Following Ref. [24], we obtain where E is the energy of the final fermion. Integrating over dE, it follows that the total decay width diverges for d < 2. This is due to the extra (1/q 2 ) factor associated with the vector propagator. However, once scale invariance is broken, values of q 2 < µ 2 are removed from the phase space: 5) and the total width remains finite and positive for d < 2. 3 Next, we discuss how scale invariance could be broken such that 1 GeV µ < M Z , first considering the influence of scalar unparticles and then assuming only vector unparticles couple to the SM.

Scalar unparticles
As pointed out in Ref. [8], scale invariance is broken by the operator where H is the SM Higgs doublet, at an energy scale , where v = 174 GeV. Having an experimentally accessible conformal window µ ≪ Λ ∼ v requires C 2 ≪ 1 [9]. Assuming µ < M Z , another upper bound on µ and C 2 can be obtained from the threshold correction to the fine structure constant [9]. If the operator exists, the value of α −1 (M Z ) remains within the current uncertainty for (3.9) Eq. (3.9) provides an upper bound on µ, whereas Eqs. (2.8, 3.3) provide a lower bound. There can be a scale invariant window below M Z between these two bounds without violating any of the other constraints discussed above. As a specific example we take d U V = 2 or 3, C V = C A = 1 and Λ = v. Setting M equal to the mono-photon bound using Eqs. (2.3, 2.6), we calculate the range of C 2 , C 4 and µ that satisfies the other constraints. As shown in Table 1 and Fig. 2, there is an allowed range of µ below M Z , provided the scalar unparticle operators couple somewhat weakly (C 4 , C 2 0.1) compared to vector operators (C V = C A = 1).

Vector unparticles
Even if only vector unparticles exist, scale invariance can still be broken if the Higgs couples to higher-dimensional operators such as O µ O µ . Furthermore, due to the higher dimensionality, the scale µ is naturally suppressed compared to the electroweak scale. 4 Consider the operators Table 1: The allowed range of µ (below M Z ) and C 2 for M at the mono-photon bound, assuming C V = C A = 1 (see Eq. (2.1)), Λ = v and C 4 = C 2 (see Eqs. (3.6, 3.8)). where we have set C 2 = C 4 = 1, and the scale dimension and Eq. (3.9) are modified as follows: As shown in Fig. 3, µ can easily lie in the allowed range.

Implications for Higgs phenomenology
The effects of scalar unparticles on Higgs phenomenology have been considered in Refs. [26,27]. 5 For scalar unparticles the same operator H † HO is responsible for breaking scale invariance and Higgs-unparticle mixing to lowest order, and thus the effects are suppressed for µ ≪ M Z . To be more explicit, the mixing between the SM Higgs boson h and the unparticle is induced by the interaction term (µ 4−d /v)Oh. Considering the effective Higgs coupling (1/v)C γγ hF µν F µν as an example, the contribution from the above interaction and Eq. (3.8) is given by [26] Provided µ ≪ M Z , this is small compared to the SM effective coupling C γγ ∼ 10 −3 . The Higgs partial decay width to fermions induced by the operatorψγ µ D µ ψO is similarly suppressed.
On the other hand, for vector unparticles scale invariance is broken by H † HO µ O µ whereas mixing is (also) induced by (3.14) Using Eq. (2.1) and Eq. (3.14), the effective fermionic operator is . (3.16) Contributions of this operator to Higgs production at a linear collider have been considered in Ref. [29]. The main effect is interference with the SM Higgs-strahlung (HZ) cross-section, which can be substantial in the e + e − → hµ + µ − , hτ + τ − , hqq channels for M close to the mono-photon bound. It is also interesting to note that the operator H † HO µ O µ induces a partial decay width

Muon pair production bounds on vector unparticles
We have already obtained a collider bound using Eq. (2.6). Other bounds can be obtained using the ratio R U ≡ σ(with unparticles)/σ(without unparticles) as well as the FBA (defined in Appendix D) for e + e − → µ + µ − , and by combining measurements at and away from the Z pole. As shown in Ref. [2] and discussed further in Ref. [10], vector couplings of unparticles will mainly affect R U away from the Z pole, and FBA at the Z pole. Axial-vector couplings have the opposite behaviour.
Due to the resonance-like behavior at µ (referred to as "un-resonance" [7]), measurements at energies around µ would be particularly sensitive to unparticle effects. Thus the bounds on c V,A (defined in Eq. (2.2)) for a given value of d will also depend on µ. As an example we plot FBA and R U for d = 1.1 in Fig. 4. Taking c A = 0.026 and c V = 0, FBA= −7.2% for √ s = 34.8 GeV if µ = 30 GeV, to be compared with −8.3% if µ ≪ 30 GeV, and −8.9% for SM. Taking into account the measurement FBA= −10.4 ± 1.3 ± 0.5% at the same center-of-mass energy [30], it is clear that the bound on c A for µ = 30 GeV will be more stringent compared to the bound for µ ≪ 30 GeV (see Fig. 5).
It should be noted that for the propagator in Eq. (3.2), the area under the un-resonance diverges for d < 1.5. However, it is likely that once scale invariance is broken, particle-like modes will appear in the spectral density [8]. For example, vacuum polarization correction from fermion loops will modify Eq. (3.2) as follows: It can therefore be expected that the unparticle will become unstable, and the area under the un-resonance will depend on the decay width. We have performed a χ 2 analysis of LEP1-Aleph, KEK-Venus and PETRA-MarkJ e + e − → µ + µ − cross-section and FBA data [30,31]. The simulation includes the vacuum polarization correction from fermion loops to the unparticle propagator (see Appendix C) and uses a fixed Z decay width Γ Z = 2.41 GeV which is the SM best-fit value for the data. Initial-state QED corrections are also included (see Appendix D).
The allowed regions in the c V -c A plane for different values of d and µ are shown in Fig. 5. The best-fit parameters and χ 2 values are listed in Table 2, and fits to FBA data with and without unparticles are displayed in Fig. 6. For values of d close to 1 where fermion-unparticle couplings are less suppressed by M 1−d Z , constraints on c V and c A are more stringent and the dependence on µ is more significant. The mono-photon bound discussed in Section 2 is stronger than the muon pair production bound for d 1.3. 6 Finally, it is worth emphasizing that the spin and scale dimension of the exchanged unparticle can be probed by analyzing the scattering angle and energy distributions of differential cross-sections in a linear collider, for both real emission and virtual exchange processes [2,4,15]. Furthermore, for polarized beams the azimuthal dependence of the final state fermion can provide an independent measure of the scaling dimension for spin-1 unparticle exchange [33].

Summary
For exact scale invariance, astrophysical and cosmological constraints are in gross conflict with the possibility of probing unparticles in colliders. We showed that for vector unparticles collider constraints become relevant only if scale invariance is broken at a scale µ 1 GeV. Breaking the scale invariance also affects collider expectations by giving rise to a resonance-like behaviour. On the other hand, unparticle effects cannot be observed at energies below the scale µ. We focused on the case 1 GeV µ < M Z which allows unparticle effects to show up in Z exchange observables, and gave demonstrations of how this can be realized through unparticle-Higgs couplings.
Simple bounds on vector unparticles have been obtained using effective contact interactions in Refs. [4,9]. Here we have made a more detailed analysis using e + e − → µ + µ − cross-section and forward-backward asymmetry data both at the Z pole and away from it, also taking into account the resonance-like behaviour associated with broken scale invariance. We found that unparticle parameters are severely constrained for values of scale dimension d close to 1. For d 1.3, constraints from mono-photon production are more stringent compared to constraints from muon pair production.

A. Unparticle contribution to the Z hadronic width
Ref. [34] studied the real and virtual massive vector boson contribution to the Z hadronic width R Z . To calculate the constraint on unparticles, we write the unparticle operator in terms of deconstructed particle fields [35]: O µ = j F j λ µ j , where the field λ µ j has mass M 2 j = j∆ 2 and In the limit ∆ → 0, the contribution to R Z is obtained by integrating the contribution from a vector boson with mass m over δ = m 2 /M 2 Z : where [34] F 1 (δ) = (1 + δ) 2 3 ln δ + (ln δ) 2 + 5(1 − δ 2 ) − 2δ ln δ x 0 dt ln(1 − t)/t is the Spence function, and uniform coupling for quarks is assumed. Note that the upper limit 1 of the δ integration is kinematic for the real emission, and the upper limit becomes ∞ for the virtual correction.
Evaluating the integrals, we obtain ∆R Z /R Z ≃ 0.01c 2 V , corresponding to a bound c V 0.3 since ∆R Z /R Z = ∆α s /π ≃ 0.001. Including the axial-vector coupling is straightforward and leads to c 0.3/ √ 2.

B. The bound from SN 1987A cooling
As discussed in Refs. [5,6,17,18,36], SN 1987A energy-loss arguments provide very restrictive constraints on unparticle couplings. In this section we discuss the constraint from pair annihilation of neutrinos and obtain the prefactor C d in Eqs. (2.8, 3.3) following the method in Refs. [35,36]. 7 The observed duration of SN 1987A neutrino burst puts a constraint on the supernova volume emissivity [37] Q 3 × 10 33 erg cm −3 s −1 , where the supernova core temperature is taken to be T SN = 30 MeV. This corresponds to As in Appendix A, we write the unparticle operator in terms of deconstructed particle fields. The cross-section for neutrino pair annihilation to λ µ j is The supernova volume emissivity is found by thermally averaging over the Fermi-Dirac distribution (see e.g. Ref. [38]): where we ignored chemical potentials (see Ref. [17]), and s = 2E 1 E 2 (1 − cos θ). The total emissivity is obtained as 8 The constraint from pair annihilation (for exact scale invariance) is discussed in Ref. [17]. The constraint from nucleon bremsstrahlung is similar in magnitude [5,6,17]. 8 See Ref. [36] for similar calculations with tensor unparticles. where We now repeat the calculation for non-zero µ. By matching to the spectral density in Eq.

C. Vacuum polarization correction
To lowest order, Π(q 2 ) in Eq. (4.1) is given as follows: is complex for the s channel with q 2 > 4m 2 f , and the imaginary part will stabilize the propagator when the real part coincides with the pole. We assume a universal coupling between the unparticle and different fermions that include charged leptons, neutrinos and quarks. A numerical example for Π(q 2 ) that is summed over the fermions is shown in Fig. 7.

D. Initial state QED corrections
Initial state QED corrections significantly affect the cross-section and FBA around µ (see Fig. 8). Since the corrections to the SM cross-section σ SM are removed from the KEK-Venus and PETRA-MarkJ data, we only consider the corrections to the unparticle exchange Again, the LEP1-Aleph data are fitted with full QED corrections. At the energy scale M Z with the scaling breaking parameter µ 1 GeV, unparticle bremsstrahlung is not effective and thus not included.