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Making the most and the best of Unparticle

Making the most and the best of Unparticle. to gain as much advantage and enjoyment as you can from sth. to a ccept a difficult situation and do as well as you can. A survey of scale invariance and conformal group. Banks-Zaks fields as an example of non-trivial IR fixed point.

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Making the most and the best of Unparticle

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

  2. 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

  3. 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.

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

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

  6. 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 … Special Conformal Transformation

  7. It is widely believed that unitary interacting scale invariant theories are always invariant under the full conformal group. (only proven in 2D) S. Coleman et al have shown that under some conditions, a scale invariant theory is also comformally invariant (including all renormalizable field theory) 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 P2. They have a continuous spectrum.

  8. 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 point Fixed Point At fixed point, scale invariance is recovered. 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?

  9. For an SU(3) gauge theory with NF massless Dirac fermion Two loop βfunction has a non-trivial zero.

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

  11. BZ Model Dimensional transmutation Scale invariant theory

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

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

  14. Unparticles have continuous spectra of masses. The usual factor for a normal field operator take time order and Fourier transformation well defined for negative P2 For non-integer dU there is a cut in the space of P2. The cut has to chosen at positive timelike P2. The cut could be seen as a combination of continuous poles.

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

  16. Interaction with SM particles through the exchange of a heavy particle of mass MU Non-renormalizable vertex With unparticle vertex and propagator, very interesting phenomenology can be studied. K. Cheung, W.Y. Keung and T.C. Yuan PRD 76 055003 C.H. Chen and C.Q Geng, PRD

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

  18. interference with SM through virtue unparticle exchange Drell-Yan Process

  19. 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.

  20. 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.

  21. 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.

  22. M. Bander, J. Feng, A. Rajaraman and Y. Shirman 0706.2677

  23. 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?

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

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

  26. 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

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

  28. 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.

  29. Vertex of two gluons and two unparticles

  30. Corresponding scalar particle pair production cross section

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

  32. 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!

  33. 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

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

  35. 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

  36. Scaling needs to scale with Mn2

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

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

  39. 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. AdS5Anti-de-Sitter Space

  40. AdS5Anti-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.

  41. Non-compact Boundary Isometry in AdS5 The metric does not change Conformal Symmetry on the boundary z=0 CFT

  42. AdS-CFT Correspondence Non-compact AdS5 model unpartcile CFT in 4D What is the dimension of the unparticle that corresponds to a massive bulk scalar field?

  43. Bulk scalar field in AdS5 Boundary operator in unparticle CFT Dimension du is the solution of Eq.

  44. Bulk scalar field correlation function in AdS5 Two point function If we care only the limit

  45. under scale transformation Isometry in AdS5

  46. Consider the compact version KK expasion For large n Mass spectrum 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)

  47. Deconstruction Non-compact AdS5 model unpartcile CFT in 4D Non-exotic, but with gravity! To localize gravity, may need a Planck brane at An ultraviolet brane corresponds to unltraviolet cutoff in 4D. Unparticle theory cutoff at RSII

  48. Gauge interaction of unparticles Non-compact AdS5 model unpartcile CFT in 4D introduce bulk color gauge field and bulk gauge transformation Bulk scalar field can be naturally colored or flavored! put SM on the brane, transforming under restricted localized gauge transformation

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

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