Making the most and the best of Unparticle to accept a difficult situation and do as well as you can to gain as much advantage and enjoyment as you can from sth

A survey of scale invariance and conformal group Banks-Zaks fields as an example of non-trivial IR fixed point Unparticle effective theory, propagator and vertex The difficulty in giving unparticle SM gauge quantum number Scale invariance breaking Gauge interaction of unparticle

It has been a long dream that the current human size is not so special. It will be business as usual after a scale transformation.

Scale Transformation The weight a unit area of bone could sustain has to change, too.

Scale transformation in QFT In high energy, the masses of all particles may be ignored and a scale invariant theory will emerge. Fundamental particle masses will break scale symmetry From studying scale invariant QFT, like massless free field, physicists found the theory is invariant under Minkowski space inversion I From translation Twe can generate anew symmetry. Special conformal transformation

Special Conformal Transformation Conformal group: Transformations that preserve the form of the metric up to a factor. It preserve the angle between two 4 vectors It includes Poincare transformations, scale transformation and …

It is widely believed that unitary interacting scale invariant theories are always invariant under the full conformal group. (only proven in 2D) The properties of a scale invariant theory are usually determined by a set of operators which is eigenfunctions of the scaling operator D. Conformal invariance severely restricts the two point function of these operators. They are not eigenfunction of mass operator P 2. They have a continuous spectrum. S. Coleman et al have shown that under some conditions, a scale invariant theory is also comformally invariant (including all renormalizable field theory)

Field theories generally exhibit scale invariant UV fixed point (often free) and scale invariant IR fixed point (often trivial, meaning non-interacting). What if the IR fixed point is non-trivial? In QFT, it is more complicated due to the presence of renormalization scale. The coupling constant depends on μ. For dimensionless g we get a dimensionful parameter Λ. Scale invariance is broken.Dimensional transmutation unless ……. at some pointFixed Point At fixed point, scale invariance is recovered.

For an SU(3) gauge theory with N F massless Dirac fermion Two loop β function has a non-trivial zero.

UV IR non-trivial IR fixed point This asymptotic free gauge theory with massless fermions has a Banks-Saks (BS) Model Scale invariance

BZ ModelDimensional transmutation Scale invariant theory

BZ Model Scale invariant theory If it becomes strong interacting near IR fixed point, massless fermions operators which are eigenfunctions of scale transformation D unparticles Integrate out degrees of freedom which is usually of order They will stay since they have continuous spectra. O U is of dimension d u

Unparticle propagator Scale invariance almost determines unparticle propagator completely. Scale invariance dictates the left scale with dimension 2 d U. This is identified as the phase space factor for n massless final particles. Unparticle with dimension d U looks like a non-integer number d U of particles.

take time order and Fourier transformation The usual factor for a normal field operator well defined for negative P 2 The cut has to chosen at positive timelike P 2. Unparticles have continuous spectra of masses. For non-integer d U there is a cut in the space of P 2. The cut could be seen as a combination of continuous poles.

This is really no surprise since unparticle has continuous mass spectrum.

Interaction with SM particles through the exchange of a heavy particle of mass M U Non-renormalizable vertex With unparticle vertex and propagator, very interesting phenomenology can be studied. K. Cheung, W.Y. Keung and T.C. Yuan PRD C.H. Chen and C.Q Geng, PRD

continuous missing energy in real unparticle emission K. Cheung, W.Y. Keung and T.C. Yuan, PRL

interference with SM through virtue unparticle exchange Drell-Yan Process

Unparticles are a hidden sector, like heavenly god They are so unlike us, normal, earthly particles. Seeing them needs to be so rare that it’s called a miracle.

To increase its importance in our earthly life and to teach us his message, God needs an incarnation, ie. becoming a human form. Unparticles needs to be given SM gauge quantum numbers. That miracle still doesn’t happen everyday means that scale invariance needs to be broken in the low energy and will manifest itself as energy gets higher.

Scale invariance most likely will be broken anyway. P. Fox, A. Rajaraman and Y. Shirman PRD 2007 The scale invariance supposedly will be broken at conformal window For the window to be not too narrow, Nonrenormalizable couplings will be suppressed. Unparticles will be unaccessible.

M. Bander, J. Feng, A. Rajaraman and Y. Shirman

However, we don’t need non-renormalizable terms to access unparticle in case they are incarnated, ie. has SM quantum number. Imagine the following scenario: Make sure we didn’t see unparticle until LHC The conformal window is about two degrees of magnitude Non-renormalizable interaction could be ignored. We ask the same question Howard asked: How does a SM flavored or colored unparticle look like in collider?

Two hurdles to overcome: How do we introduce scale invariance breaking effects? How do we flavor or color a unparticle?

How do we introduce scale invariance breaking effects? Parameterize the breaking with an infrared cutoff.

Parameterize the breaking with an infrared cutoff m. It reduced to Georgi’s unparticle propagator as and reduced to particle propagator with mass m as

How do we flavor or color a unparticle? Gauge interaction of unparticles

How do we flavor or color a unparticle? Gauge interaction of unparticles The unparticle propagator naively will imply a non-local Lagrangian: For gauge symmetry, insert a Wilson Line Vertex of a Gluon coupled to two unparticles.

Vertex of two gluons and two unparticles

Corresponding scalar particle pair production cross section

Option II My suggestion is to use the representation of unparticle as bulk field in extra dimensional model

The unparticle propagator contains a cut for timelike P. A cut line can be decomposed into a collection of point poles with the gap goes to zero! An unparticle may correspond to a collection of particles!

Scale invariance It suggests a collection of non-interacting particles created by operator O with continuous mass distribution. Start with a discrete form: mass gap Scaling

Kaluza-Klein Modes of Bulk Field Bulk Fields contains Kaluza-Klein (KK) States Ψ n with wavefunctions: due to periodicity. The extra-dimension wave number E-p relation KK states look like having 4D masses ……. Towers of particles appear in extra dimensional models.

Kaluza-Klein Modes of Bulk Field ……. Non-compact extra dimensions gives towers of particles with continuous mass distribution. Assume only deconstructed unparticle see the non-compact extra dimension. Even gravity won’t see it. Or the gravity is localized! RSII

needs to scale with M n 2 Scaling

Assume that density of states For m extra dimensions Density of states is proportional to the hyper sphere shell: ADD realization W.Y. Keung Consider ADD with m extra dimensions and a bulk scalar field is a KK state

unparticle with dimension 3/2Bulk scalar field in 5 non- compact dimensions need a ultraviolet cutoff Bulk scalar field in 4 non-compact continuous dimensions and one non- compact discrete dimension ADD realization of deconstruction

ADS-CFT Consider one extra dimension: z Need 4D Poincare symmetry: 4D unparticle theory is scale invariant In 5D, no longer conformal, it needs to corresponds to an isometry of the metric. AdS 5 Anti-de-Sitter Space

AdS 5 Anti-de-Sitter Space: the most symmetric spacetime with negative curvature Conformally Flat frame It’s common to take out the dimensions from all the coordinates.

Boundary Non-compact Isometry in AdS 5 The metric does not change Conformal Symmetry on the boundary z=0 CFT

AdS-CFT Correspondence unpartcile CFT in 4DNon-compact AdS 5 model What is the dimension of the unparticle that corresponds to a massive bulk scalar field?

Bulk scalar field in AdS 5 Boundary operator in unparticle CFT Dimension d u is the solution of Eq.

Bulk scalar field correlation function in AdS 5 Two point function If we care only the limit

under scale transformation Isometry in AdS 5

Consider the compact version For large n Mass spectrum KK expasion Same thing can be done for other Lorentz structure (there is a table of relations between dimensions and bulk masses in the Maldacena et al review)

unpartcile CFT in 4DNon-compact AdS 5 model Deconstruction Non-exotic, but with gravity! To localize gravity, may need a Planck brane at An ultraviolet brane corresponds to unltraviolet cutoff in 4D. RSII Unparticle theory cutoff at

unpartcile CFT in 4DNon-compact AdS 5 model Gauge interaction of unparticles introduce bulk color gauge field and bulk gauge transformation put SM on the brane, transforming under restricted localized gauge transformation Bulk scalar field can be naturally colored or flavored!

IR brane introduce a natural infrared cutoff An IR brane will naturally give an IR cutoff and hence a breaking of scale invariance a bonus

Other deconstruction ( flat without gravity ) Fractal Theory SpaceFractional Dimension