1. A two-loop calculation in quantum field theory on orbifolds Nobuhiro Uekusa

2. Description of physical quantities Action principle Virtual processes by quantum loop corrections In 4D theory 2 LHC erahigher energies

3. Description of physical quantities Action principle Virtual processes by quantum loop corrections Invariance of theory Conserved currents In 4D theory 3 LHC erahigher energies

4. Renormalizability Finite number of interactions New counterterms not required 4 accurate prediction

5. Renormalizability Finite number of interactions New counterterms not required Invariance of theory does not forbid non-renormalizable interactions A non-renormalizable interaction New counterterms 5 accurate prediction

6. Renormalizability Requirement in addition to invariance of theory? 6

7. Renormalizability Requirement in addition to invariance of theory? Not compulsory Irrelevant operators Negligible contributions to physical quantities In 4D, usually non-renormalizable interactions are supposed to be suppressed by a UV cutoff of a theory. 7

8. Renormalizability Requirement in addition to invariance of theory? Not compulsory Irrelevant operators Negligible contributions to physical quantities In 4D, usually non-renormalizable interactions are supposed to be suppressed by a UV cutoff of a theory. Effective theory with a large cutoff can be predictable without requiring renormalizability Only ? 8 in 4D

9. renormalizable and non-renormalizable interactions coexist. fields as 4D modes can have dim-4 operators In 4D In a theory with compactifed extra dim Simliar to renormalizable terms in 4D If coefficients of other operators are small, such a theory might be predictable with a certain accuracy. 9

10. 10 The coefficients of higher- dimensional operators Unknown Should be eventually determined fields as 4D modes can have dim-4 operators

11. 11 Some attitudes Try to construct a consistent theory to specify all the non-renormalizable interactions Search for rules or orders for possible interactions at each given loop level The coefficients of higher- dimensional operators Unknown Should be eventually determined

12. 12 Search for rules or orders for possible interactions at each given loop level Quantum loop corrections to 2-point functions in 5D theory on orbifold S /Z 2 1

13. 13 The action for the real scalar field The boundary conditions for Possible Lagrangian counterterms and

14. 14 Mass term No wave function Mass term No wave function Mass term Wave function

15. 15 Mass term No wave function Mass term No wave function Mass term Wave function

16. 16 1-loop KK mode expansion Sum of diagrams for KK modes Momentum integrals Dimensionless

17. 17 1-loop KK mode expansion Sum of diagrams for KK modes Momentum integrals 0 0 0 0 2n n ff+2n n ff n Internal mode indep of external mode

18. 18 1-loop KK mode expansion Sum of diagrams for KK modes Momentum integrals Boundary terms Bulk terms

19. 19 Fractions Integral expression of Gamma function 2-loop KK mode sum Poisson’s summation Divergent part momentum integral with a Calculation method cutoff regularization counterterm

20. 20 Now (p ) divergence has been found It needs to be taken into account in the starting action integral 2 Toward extraction of physical quantities without requiring renormalizablility

21. 21 An effect of higher terms Take into account (p ) terms 2 Equation of motion (Fourier transformed) parameter Propagator

22. 22 An effect of higher terms Propagator Two poles Unusual sign Decaying mode

23. 23 An effect of higher terms Two poles Unusual sign Decaying mode Propagator

24. 24 An effect of higher terms as a loop effect Unnatural degree with a mass larger than the cutoff The correction is extracted with a tuning as in 4D large Propagator

25. 25 Even higher loop 4-loop, (p ) corrections 2 3 3 poles in propagator p p k1 k2 k3 k4 K1+k2+k3 P-k1-k2-k3-k4 P-k1-k2

26. 26 1 Quantum loop corrections to 2-point functions in 5D theory on orbifold S /Z 2 2-loop, (p ) div 2 4-loop, (p ) div 2 3 For extraction of corrections for 2-pt function, the UV cutoff needs to be orders of magnitude larger compared to the compactified scale This behavior is in agreement with the conventional observation with that contributions of higher dim operators are small for a large cutoff. Two or more poles in propagater SUMMARY

27. 27 Evaluation of bulk and boundary terms Mode expansion Boundary terms have off-diagonal components wrt n On the other hand, bulk terms are diagonal wrt n

28. 28 Evaluation of divergence Fractions Integral expression of Gamma function KK mode sum Poisson’s summation Intuitive interpretation of bulk divergence e.g.

29. 29 Evaluation of divergence

30. 30 Evaluation of divergence

31. 31 1 Quantum loop corrections to 2-point functions in 5D theory on orbifold S /Z 2 2-loop, (p ) div 2 4-loop, (p ) div 2 3 For extraction of corrections for 2-pt function, the UV cutoff needs to be orders of magnitude larger compared to the compactified scale This behavior is in agreement with the conventional observation with that contributions of higher dim operators are small for a large cutoff. Two or more poles in propagater SUMMARY