Standard Model And High-Energy Lorentz Violation,

1. Standard Model AndHigh-Energy Lorentz Violation, Damiano Anselmi 中国科学院理论物理研究所 北京 2010年4月6日

2. based on the hep-ph papers [0a] D.A., Standard Model Without Elementary Scalars And High Energy Lorentz Violation, Eur. Phys. J. C 65 (2010) 523 and arXiv:0904.1849 [hep-ph] [0b] D.A., Weighted power counting, neutrino masses and Lorentz violating extensions of the Standard Model, Phys. Rev. D 79 (2009) 025017 andarXiv:0808.3475 [hep-ph] [0c] D.A. and M. Taiuti, Renormalization of high-energy Lorentz violating QED, Phys. Rev. D in press, arxiv:0912.0113 [hep-ph] [0d] D.A. and E. Ciuffoli, Renormalization of high-energy Lorentz violating four fermion models, Phys. Rev. D in press, arXiv:1002.2704[hep-ph]

3. based on the hep-ph papers [0a] D.A., Standard Model Without Elementary Scalars And High Energy Lorentz Violation, Eur. Phys. J. C 65 (2010) 523 and arXiv:0904.1849 [hep-ph] [0b] D.A., Weighted power counting, neutrino masses and Lorentz violating extensions of the Standard Model, Phys. Rev. D 79 (2009) 025017 andarXiv:0808.3475 [hep-ph] [0c] D.A. and M. Taiuti, Renormalization of high-energy Lorentz violating QED, Phys. Rev. D in press, arxiv:0912.0113 [hep-ph] [0d] D.A. and E. Ciuffoli, Renormalization of high-energy Lorentz violating four fermion models, Phys. Rev. D in press, arXiv:1002.2704[hep-ph] and previous hep-th papers [1] D.A. and M. Halat, Renormalization of Lorentz violating theories, Phys. Rev. D 76 (2007) 125011 andarxiv:0707.2480 [hep-th] [2] D.A., Weighted scale invariant quantum field theories, JHEP 02 (2008) 05 and arxiv:0801.1216 [hep-th] [3] D.A., Weighted power counting and Lorentz violating gauge theories. I: General properties, Ann. Phys. 324 (2009) 874  and arXiv:0808.3470 [hep-th] [4] D.A., Weighted power counting and Lorentz violating gauge theories. II: Classification, Ann. Phys. 324 (2009) 1058 and arXiv:0808.3474 [hep-th]

4. Lorentz symmetry is a basic ingredient of the Standard Model of particles physics.

5. Lorentz symmetry is a basic ingredient of the Standard Model of particles physics. However, several authors have argued that at high energies Lorentz symmetry and possibly CPT could be broken

6. Lorentz symmetry is a basic ingredient of the Standard Model of particles physics. However, several authors have argued that at high energies Lorentz symmetry and possibly CPT could be broken. The Lorentz violating parameters of the Standard Model (Colladay-Kostelecky) extended in the power-counting renormalizable sector have been measured with great precision.

7. Lorentz symmetry is a basic ingredient of the Standard Model of particles physics. However, several authors have argued that at high energies Lorentz symmetry and possibly CPT could be broken. The Lorentz violating parameters of the Standard Model (Colladay-Kostelecky) extended in the power-counting renormalizable sector have been measured with great precision. It turns out that Lorentz symmetry is a very precise symmetry of Nature, at least in low-energy domain.

8. Lorentz symmetry is a basic ingredient of the Standard Model of particles physics. However, several authors have argued that at high energies Lorentz symmetry and possibly CPT could be broken. The Lorentz violating parameters of the Standard Model (Colladay-Kostelecky) extended in the power-counting renormalizable sector have been measured with great precision. It turns out that Lorentz symmetry is a very precise symmetry of Nature, at least in low-energy domain. Several (dimensionless) parameters have bounds

9. Why is it interesting to consider quantum field theories where Lorentz symmetry is explicitly broken?

10. Why is it interesting to consider quantum field theories where Lorentz symmetry is explicitly broken? The set of power-counting renormalizable theories is considerably “small”

11. Why is it interesting to consider quantum field theories where Lorentz symmetry is explicitly broken? The set of power-counting renormalizable theories is considerably “small” Relaxing some assumptions can enlarge it, but often it enlarges it too much

12. Why is it interesting to consider quantum field theories where Lorentz symmetry is explicitly broken? The set of power-counting renormalizable theories is considerably “small” Relaxing some assumptions can enlarge it, but often it enlarges it too much Without locality in principle every theory can be made finite

13. Why is it interesting to consider quantum field theories where Lorentz symmetry is explicitly broken? The set of power-counting renormalizable theories is considerably “small” Relaxing some assumptions can enlarge it, but often it enlarges it too much Without locality in principle every theory can be made finite Without unitarity even gravity can be renormalized

14. Why is it interesting to consider quantum field theories where Lorentz symmetry is explicitly broken? The set of power-counting renormalizable theories is considerably “small” Relaxing some assumptions can enlarge it, but often it enlarges it too much Without locality in principle every theory can be made finite Without unitarity even gravity can be renormalized Relaxing Lorentz invariance appears to be interesting in its own right

15. Why is it interesting to consider quantum field theories where Lorentz symmetry is explicitly broken? The set of power-counting renormalizable theories is considerably “small” Relaxing some assumptions can enlarge it, but often it enlarges it too much Without locality in principle every theory can be made finite Without unitarity even gravity can be renormalized Relaxing Lorentz invariance appears to be interesting in its own right It could be useful to define the ultraviolet limit of quantum gravity and study extensions of the Standard Model

16. Why is it interesting to consider quantum field theories where Lorentz symmetry is explicitly broken? The set of power-counting renormalizable theories is considerably “small” Relaxing some assumptions can enlarge it, but often it enlarges it too much Without locality in principle every theory can be made finite Without unitarity even gravity can be renormalized Relaxing Lorentz invariance appears to be interesting in its own right It could be useful to define the ultraviolet limit of quantum gravity and study extensions of the Standard Model Here we are interested in the renormalization of Lorentz violating theories obtained improving the behavior of propagators with the help of higher space derivatives and study under which conditions no higher time derivatives are turned on to be consistent withunitarity

17. Why is it interesting to consider quantum field theories where Lorentz symmetry is explicitly broken? The set of power-counting renormalizable theories is considerably “small” Relaxing some assumptions can enlarge it, but often it enlarges it too much Without locality in principle every theory can be made finite Without unitarity even gravity can be renormalized Relaxing Lorentz invariance appears to be interesting in its own right It could be useful to define the ultraviolet limit of quantum gravity and study extensions of the Standard Model Here we are interested in the renormalization of Lorentz violating theories obtained improving the behavior of propagators with the help of higher space derivatives and study under which conditions no higher time derivatives are turned on to be consistent withunitarity The approach that I formulate is based of a modified criterion of power counting, dubbed weighted power counting

18. We may assume that there exists an energy range that is well described by a Lorentz violating, but CPT invariant quantum field theory. If the neutrino mass has the Lorentz violating origin we propose, then and the mentioned range spans at least 4-5 orders of magnitude.

19. Scalar fields Break spacetime in two pieces: Breakcoordinates and momenta correspondingly: Consider the free theory This free theory is invariant under the “weighted” scale transformation is the “weighted dimension”

20. The propagator behaves better than usual in the barred directions Adding “weighted relevant’’ terms we get a freetheory that flows to the previous one in the UV and to the Lorentz invariant free theory in the infrared (actually, the IR Lorentz recovery is much more subtle, see below)

21. Add vertices constructed with , and . Call their degrees under N = number of legs, = extra label Other quadratic terms can be treated as “vertices” for the purposes of renormalization

22. Consider a diagram G with L loops, I internal legs, E external legs and vertices of type (N , ) is a weighted measure of degree Performing a“weighted rescaling” of external momenta, together with a change of variables we see that is a homogeneous weighted function of degree Its overall divergent part is a homogeneous weighted polynomial of degree

23. Using the standard relations we get Where Renormalizable theories have Indeed implies

24. Writing we see that polynomiality demands and the maximal number of legs is Conclusion: renormalization does not turn on higher time derivatives E = 2 implies 2 E > 2 implies < 2

25. Examples n=2: six-dimensional -theory Strictly-renormalizable models are classically weighted scale invariant, namely invariant under The weighted scale invariance is anomalous at the quantum level

26. Four dimensional examples n = 2 n = 2 n = 3

27. Källen-Lehman representation and unitarity Cutting rules

28. Causality Our theories satisfy Bogoliubov's definition of causality which is a simple consequence of the largest time equation and the cutting rules For the two-point function this is just the statement if >0 immediate consequence of

29. Fermions The extension to fermions is straightforward. The free lagrangian is An example is the four fermion theory with An example of four dimensional scalar-fermion theory is

30. High-energy Lorentz violating QED

31. High-energy Lorentz violating QED Gauge symmetry is unmodified

32. High-energy Lorentz violating QED Gauge symmetry is unmodified A convenient gauge-fixing lagrangian is

33. Integrating the auxiliary field B away we find

34. Integrating the auxiliary field B away we find Propagators

35. Integrating the auxiliary field B away we find Propagators This gauge exhibits the renormalizability of the theory, but not its unitarity

36. Coulomb gauge

37. Coulomb gauge

38. Coulomb gauge Two degrees of freedom with dispersion relation

39. Coulomb gauge Two degrees of freedom with dispersion relation The Coulomb gauge exhibits the unitarity of the theory, but not its renormalizability

40. Coulomb gauge Two degrees of freedom with dispersion relation The Coulomb gauge exhibits the unitarity of the theory, but not its renormalizability Correlation functions of gauge invariant objects are both unitary and renormalizable

41. Weighted power counting

42. Weighted power counting The theory is super-renormalizable Counterterms are just one- and two-loops

43. Weighted power counting The theory is super-renormalizable Counterterms are just one- and two-loops

44. High-energy one-loop renormalization

45. High-energy one-loop renormalization

46. High-energy one-loop renormalization

47. Low-energy renormalization

48. Low-energy renormalization

49. Low-energy renormalization Two cut-offs, with the identification

50. Low-energy renormalization Two cut-offs, with the identification Logarithmic divergences give