Unparticles are first discussed by Howard Georgi in 2007 [2]. The idea is based on scale invariance. In 1981, Banks and Zaks [4] observed that a QCD like theory with more than six flavors could have a nontrivial infrared fixed point. That means the theory will become scale invariant at a finite energy.
Scale invariance is a powerful concept that has wide applications in many different disciplines of physics. Banks and Zaks combined information with strong coupling and large-NF massless fermions gauge theories. In string theory, scale invariance plays an even more fundamental role since it is part of the local diffeomorphism X Weyl reparametrization invariance group of the two-dimensional Riemann surfaces[4].
However, at the low energy world of particle physics, what we observe is a plethora of elementary and composite particles with a wide spectrum of masses [5]. In quantum field theory, the dilatation generator D for scale transformation does not commute with the spacetime translation generators Pμ, if we make a coordinate transformation xμ →λxμ, the 4-momentum will transform as Pμ →λ−1Pμ, and the field will transform as φ →λ−dφ, where the dimension d depends on the field.
However, even a massless theory, is more complicated due to the presence of a renormalization scale μ. And because the coupling constant g depends on μ, for dimensionless g, we need a dimensionful parameter Λ to eliminate the dimension of μ.
Hence the scale invariance is broken because of the nonrenormalizable parameter, unless there exist a fixed point which makes 0
ln =β = μ d
dg . Figure 1 shows this point, the UV fixed point in QCD, when μ→∞⇒β →0:
Figure 1: UV fixed point in QCD
Banks and Zakes concluded that a large class of continuum models with fermions in real representations of the gauge group do not have spontaneous chiral symmetry breaking. Hence the theory will be dictated by the scale invariance principle. Figure 2 shows the asymptotic free gauge theory with massless fermions has a non-trivial IR fixed point:
Figure 2:Scale invariance, when μ↓⇒β →0,g →g*
Georgi, motivated by the Banks-Zaks theory, suggested that a scale invariant sector with a nontrivial infrared fixed-point behaves rather peculiar from the perspective of particle physics. Such a theory in low energy is dominated by a set of operators with anomalous (unconventional) dimensions. Most of the degree of freedom will be strong interacting above the scale μ = ΛU. If it becomes strong interacting near IR fixed point, they will disappear (and can be integrated out) in the low energy theory.
Hence at low energy we can adopt an effective theory with these operators replaced by low energy effective field operators-denoted as unparticles, since they have continuous spectra, and all the other degrees of freedoms can be integrated out. By definition the unparticle exhibits continuous mass spectrum stretching from zero to infinity.
This would be a problem if it interacts strongly with Standard model particles, we will discuss about this point later. Now, the two sets are hence designed to interact through exchange a hidden particle with large mass Mu. The nonrenormalizable couplings involving both SM and BZ fields are suppressed by Mu. These interactions
UV IR
g*
have the generic form
where OSM are operators with mass dimension dSM built from SM fields and OBZ are operators with mass dimension dBZ built from BZ fields, i.e. hidden sector fields, and k = dBZ + dSM – 4. Cu is dimensionless coefficient function. Scale invariance in the hidden sector emerges at the energy scale Λu where the infrared fixed point appears.
In the effective theory below Λu the interactions of Eq. (2.1) take the form
u where du is the scaling dimension of the unparticle operator Ou.
Since the low energy theory is scale invariant, the physics of the unparticle operators Ou will be dictated by scale invariance. In Ref. [2], Georgi showed by scale invariance the propagator of Ou can be totally determined up to a constant. He started with the vacuum matrix element:
( ) [ ] ( ) ( )
where the constant is irrelevant for physics and will be fixed as
)
π for the reason in phase space calculations.
Then take time order and Fourier transformation, we can get
( ) ( )
space of P2, which has to be chosen at positive timelike P2 and could be seen as a combination of continuous poles.
The unparticle, with a continuous mass spectrum, is very different from ordinary particle. Their effects might be detectable in missing energy distributions and interference with SM amplitudes [2, 6, 8, 14]. In missing energy observation, the unparticle phase space will look like fractional numbers of final massless particles.
And due to the peculiar behavior of its propagator, the virtual exchange of an unparticle will interfere strongly with SM contributions. The beauty of the theory is that all of these effects can be deduced solely from the scale invariance of the unparticle sector without knowing the details of the theory.
The muon pair production in electron-positron collision e++e− →μ+ +μ−is a beautiful example where the peculiar property of the unparticle propagator will show up [5]. The muon pair could be generated either through virtual photon, Z boson or unparticle production. As Georgi observed in his paper [2], the unparticle propagator contains both a real part and an imaginary part along the whole momentum range and hence it will interfere with Z boson and photon propagator even at Z pole, where the Z boson is dominated by its imaginary part. This point is brought home by calculating the fractional change in the total cross section.
Figure 3.
Assume that the interaction between leptons and a vector unparticle can be divided into a vector part and axial-vector part:
μ μ
μ μ γ γ
γ k u
u d k
u AU k u
u d k
u
VU l lO
M O C
l M l
C u u
5 1
1− +−
+ Λ
Λ + (2.5)
For the convenience of discussion, we can define the couplings as:
u u u
u
d Z k u
d k
u AU d AU
Z k u
d k
u VU
VU M M
C C M
M
C C −
− +
−
−
+ Λ
Λ =
= 11 11 (2.6) The case of pure axial-vector coupling cVU =0 is the most illustrative. In this case, interference only appears between virtual unparticle and virtual Z boson, since photon only couples vectorially. An unparticle propagator
( )
( 2 ) 2sin 1 2
− −
− U
U d
U
d P i
d
iA ε
π
contains a phase φU =−(dU −1)π along the whole momentum range. Hence it interferes with Z propagator everywhere and the leading contribution is of order
2
cAU instead of cAU 4, we show it in Figure 4.
The effect is sensitive to the unparticle dimension. The real part of Eq. (2.4) is positive for 1<du <1.5 and negative for 1.5<du <2 and the real part of Z propagator is negative below Z pole and positive above Z pole. Hence the interference is destructive (constructive) below (above) the pole for 1<du <1.5 and vice-versa for 21.5<du < . This interesting behavior can be seen from the diagram above.
In the case of pure vector coupling cAU =0, interference between virtual unparticle and virtual Z boson is small. The interference with photon dominates. The photon propagator is real and always positive. So we expect the interference is constructive for 1<du <1.5 and destructive for 1.5<du <2, as show in Figure 5.