1. Particle Interactions with Modified Dispersion Relations Yi Ling （凌意） IHEP,CAS & Nanchang University 2012 两岸粒子物理与宇宙学研讨会, 重庆,05/09/2012

2. Outlines The fate of Lorentz symmetry at Planck scale Introduction to deformed special relativity What is the difference between Lorentz violation theory and deformed special relativity The composition law of particles with modified dispersion relations

3. The fate of Lorentz symmetry at Planck scale “Planck length paradox” ： Lorentz contraction in special relativity: Energy-momentum relation: The existence of the minimal length that can be measured

4. The fate of Lorentz symmetry at Planck scale The possibilities of Lorentz symmetry at high energy level 1.Keeping the original form 2.Manifestly broken 3.Deformed Deformed special relativity (Doubly special relativity) (DSR) was originally proposed in the context of quantum gravity phenomenology to reconcile the relativity principle and the existence of the minimal length scale which is uniform and invariant to all observers.

5. The relativity of inertial frames, two universal constants: 1) In the limit, the speed of a photon goes to a universal constant,. 2) in the above condition is also a universal constant. As a result, the energy-momentum relation is usually modified to Introduction to deformed special relativity

6. Lorentz transformation in standard special relativity (1+1 dim.) Lorentz transformation in deformed special relativity e.g.

7. Introduction to deformed special relativity Lorentz transformation in deformed special relativity

8. Introduction to deformed special relativity Remark: the Lorentz transformation law depends on the form of modified dispersion relation

9. Two open problems in DSR Usually the standard dispersion relation in special relativity will be modified with correction terms However, such a modification in theory would lead to some severe problems…

10. Two open problems in DSR A field theory with MDR is still absent. How to define the position space? ?

11. Two open problems in DSR The soccer problem This modified dispersion relation is not applicable to composite particles and macroscopic objects. Thus it is not universal but particle number dependent.

12. the difference between Lorentz violation theory and deformed special relativity Example: the derivation of the threshold value of the interaction A.In standard special relativity Center-of-mass reference frame: Laboratory reference frame:

13. the difference between Lorentz violation theory and deformed special relativity B.In Lorentz violation theory special relativity Center-of-mass reference frame: No sense Laboratory reference frame:

14. the difference between Lorentz violation theory and deformed special relativity e.g.

15. the difference between Lorentz violation theory and deformed special relativity C.In deformed special relativity Center-of-mass reference frame: the same result can be obtained Laboratory reference frame: I.

16. the difference between Lorentz violation theory and deformed special relativity II.

17. The composition law in special relativity revisited Consider two elementary particles which may have different masses We define a composite particle through a process in which the covariant momentum is conserved An invariant quantity of the composite particle is (1+1 dim.)

18. The composition law in SR revisited Some remarks: If we define, we still have In general They are equal if and only if In general, the composite particle could not be elementary. Universal M

19. The composition law in SR revisited It is straightforward to extend it to the composition of many particles: If we define, we still have In general

20. The composition law in SR revisited The transformation law of the energy and momentum under the Lorentz boost in 1+1 space time Thus It is easy to check that for a composite particle

21. The composition law in SR revisited An interaction involving n incoming particles and m outgoing particles The conservation law of momentum is preserved under the Lorentz boost in the sense that

22. The composition law in DSR Consider an elementary particle with a modified dispersion relations as Obviously, it is not an invariant quantity under the standard Lorentz boost In DSR, a deformed boost generator is proposed so as to preserve it to be an invariant quantity up to the first order correction of the Planck length.

23. The composition law in DSR However, such a choice is not unique. An alternative deformation When consider the composition law of particles, one need look for some specific laws of composition of momenta which are supposed to be compatible with the deformed boosts one has chosen. And in general, such choices would unavoidably lead to the relative-locality of the space of momenta.

24. The composition law in DSR Our central goal in this talk We intend to argue that if we input some rules on picking up one specific form for deformed boost among all the possible choices, then the relative-locality of the space of momenta may be avoided.

25. The composition law in DSR We introduce a notion of effective momentum

26. The composition law in DSR We propose a composition law for two elementary particles Remark: it is interesting to show that if and only if

27. The composition law in DSR One can easily check the following identities for a composite particle

28. The composition law in DSR It can be further written into a compact form which depends on the number of particles manifestly given that 1. 2. In the relativistic limit, 3. In the non-relativistic limit,

29. The composition law in DSR One can easily check

30. The composition law in DSR Extension to arbitrary composite particle or macroscopic object which is composed of n elementary particles

31. The composition law in DSR For a macroscopic object with n particles in thermal equilibrium, it is reasonable to assume that then

32. The composition law in DSR A general interaction

33. The composition law for many sorts of elementary particles Two elementary particles with different dispersion relations We introduce a notion of effective energy (I)(II)

34. The composition law for many sorts of elementary particles One can easily check that The invariant quantity for the composite particle

35. The composition law for many sorts of elementary particles It can be further written into a compact form which depends on the number of particles manifestly if 1. In the relativistic limit 2. In an equilibrium state

36. The composition law for many sorts of elementary particles A composite particle which contains n elementary particles with dispersion relation (I) and m elementary particles with dispersion relation (II)

37. the difference between Lorentz violation theory and deformed special relativity General modified dispersion relations A point of view from rainbow spacetime

38. Summary We propose a composition law of momenta for a multi- particle system in deformed special relativity. The form of modified dispersion relation for a composite particle or macroscopic object is not universal but dependent on the number of elementary particles it consists of. We introduce a notion of effective energy and momentum for particles such that a specific deformed Lorentz boost generator can be constructed. The benefits of such deformed Lorentz boosts are twofold. i) A composition law of momenta compatible with the deformed Lorentz boost can be defined without introducing a notion of relative-locality of the space of momenta. ii) We provide a specific law of composition of momenta for interactions involving non-universal dispersion relations such that the invariance of the conservation law under the deformed Lorentz boost can be easily achieved.