1. Reevaluation of Neutron Electric Dipole Moment with QCD Sum Rules Natsumi Nagata Nagoya University National Taiwan University 5 November, 2012 J. Hisano, J. Y. Lee, N. Nagata, and Y. Shimizu, Phys. Rev. D85 (2012) 114044.

2. Outline 1.Introduction 2.QCD Sum Rules (review) 3.NEDM with QCD Sum Rules 4.Conclusion & Prospects

3. 1. Introduction

4. Neutron Electric Dipole Moment (nEDM) This interaction violates parity (P) and time-reversal (T) invariance. Hamiltonian (non-relativistic) Lagrangian CP (CPT theorem)

5. Neutron Electric Dipole Moment (nEDM)

6. nEDM from CKM matrix in the Standard Model CKM matrix source of CP-violation in the SM Its contribution to the nEDM is quite suppressed because it is flavor-changing interaction. I. B. Khriplovich and A. R. Zhitnitsky (1982), T. Mannel and N. Uraltsev (2012). Experimental limit (Institut Laue-Langevin) Phys.Rev.Lett. 97, 131801 (2006). The nEDM provides a clean, background-free probe of the CP- violating interactions in physics beyond the SM.

7. Effective Lagrangian CP-violating parameters quark EDMsquark CEDMs θ term (QCD scale; up to dimension 5) (physical parameter)

8. Goal  We use the method of QCD Sum Rules  Some QCD parameters are extracted from the results obtained with the lattice simulations Results We derive a conservative limit for the contributions of the CP violating operators compared with the previous calculations. Experimental limit on nEDM Experimental limit on nEDM Limits on CP @ parton-level Limits on CP @ parton-level

9. 2. QCD Sum Rules

10. QCD Sum Rules M. Shifman, A. Vainshtein, V. Zakharov, Nucl. Phys. B147, 385 (1979) M. Shifman, A. Vainshtein, V. Zakharov, Nucl. Phys. B147, 448 (1979) The correlator of the hadron fields is evaluated in terms of Operator Product Expansion (OPE) By using the dispersion relations, it is related with the sum of the contributions of the hadronic states. Short-distance contribution Long-distance contribution Perturbative QCD VEVs of quark/gluon operators

11. Correlation function Spectral function j(x): a hadron field Spectral representation

12. Dispersion relation The Cauchy formula Let us connect the region of q 2 0 (long-distance, physical) Choosing the contour in the right figure, follows. Here, we use z

13. Since the above expression has ultraviolet divergence, we subtract from Π(q 2 ) the first few terms of its Taylor expansion @ q 2 = 0. In the case of q 2 << 0, one can calculate it perturbatively (quark picture) It encodes information of Hadron spectrum However, this expression is inconvenient due to the presence of unknown subtraction terms. Moreover, little is known about the spectral function. Dispersion relation

14. Borel transformation eliminates the subtraction terms in the dispersion relation exponentially suppresses the contributions from excited resonances and the continuum states Borel transformation M: Borel mass

15. QCD sum rules By applying the Borel transformation to the dispersion relation, We obtain, This equation is so-called QCD sum rules

16. Operator product expansion (OPE) Π(q 2 ) (q 2 < 0) is evaluated in terms of the operator product expansion (OPE). Here we deal with the long- & short-distance contribution separately. Short-distance contribution included in the Wilson coefficients C i (Q 2 ) Long-distance contribution included in the VEVs (condensates)

17. Hadron contribution Im Π(s) ＝ Contribution of one-particle state (pole) + excited/continuum states (branch cut) The former has information we want to extract The latter is suppressed by the Borel transformation, but often causes theoretical uncertainty. Excited/continuum states  Quark-hadron duality  Appropriate model

18. 3. NEDM with QCD Sum Rules

19. Effective Lagrangian CP-violating parameters quark EDMsquark CEDMs θ term (QCD scale; up to dimension 5) (physical parameter)

20. Calculation Previous work M. Pospelov and A. Ritz (1999, 2001) The correlator of neutron currents is evaluated in two ways: (i) Operator product expansion (ii) Phenomenological model Connect after Borel transformation Connect after Borel transformation Short-distance Long-distance … nEDM d n

21. Correlation function of the neutron fields Correlation function We extract the nEDM from the correlator. Extra phase factor α n mixes the nEDM with the magnetic dipole moments. We focus on the chiral invariant term Neutron field: η n (x)

22. Neutron interpolating field We find that β = 1 is an optimal choice because it suppresses the higher-order contribution removes the mixing effects of currents In order to evaluate the correlator, we need to express the neutron field as a composite operator of quark fields with the same quantum numbers as those of neutron.

23. Phenomenological calculation N NN N ＊ N ＊ N ＊ Double pole Single pole No pole We assume A: const. and B 〜 0 in the following calculation

24. OPE calculation We carry out the calculation up to the N.L.O.

25. OPE result Λ: an arbitrary parameter with mass-dimension 1. Θ is a linear combination of the CP-violating parameters. χ, κ, … are the QCD parameters determined elswhere.

26. QCD Sum Rule Now we connect the two results after the Borel transformation M: Borel mass Borel mass dependence of the r.h.s. One can pick out nEDM from the tangent line to the function shown in the left figure.

27. Error estimate The sum rule gives the values of d n and A at a given Borel Mass M. Central value of d n is determined where the Double pole contribution is dominant. We estimate the error of the calculation by requiring that the Single pole contribution is less than 30% of the Double pole contribution. Ratio of Single and Double pole contributions

28. λ n (Lattice results) Y. Aoki et. al. (2008) In previous work, λ n is also evaluated by using the QCD sum rules. The lattice QCD value is several times larger than that evaluated based on the QCD sum rules. D. B. Leinweber (1997) The resultant nEDM value is smaller than those in the previous literature. We extract the value of λ n from the lattice simulations.

29. Results (phen) (OPE) (Lattice) By substituting the QCD parameters, we obtain This result is about 70% smaller than previous results. It gives a conservative limit for the contributions of the CP-violating operators. J. Hisano, J. Y. Lee, N.N., Y. Shimizu (2012)

30. 4. Conclusion & Prospects

31. Chiral perturbation theory R.J. Crewther, P. Di Vecchia, G. Veneziano, E. Witten (1979) A. Pich, E. de Rafael (1991) J. Hisano, Y. Shimizu (2004) Strange content of nucleon Unknown matrix elements

32. Future prospects Hadron-loop calculation is carried out in the chiral perturbation theory. CP-odd meson-baryon couplings are evaluated with QCD sum rules. Lattice results are used for the QCD parameters K. Fuyuto, J. Hisano, N.N., in preparation.

33. Conclusion  We have evaluated the nEDM based on the method of the QCD sum rules.  By using input parameters obtained from the lattice simulation, we have derived a conservative limit for the contributions of the CP-violating operators.

34. Backup

35. Chiral rotation By using the ciral transformations, we move to a convenient basis. U(1) A transformation Θ-term is rotated into γ 5 -mass term. SU(3) A transformation The vacuum is aligned in a ``good” direction. Tadpoles for pseudo-scalar mesons should vanish.

36. Effective Lagrangian

37. Peccei-Quinn mechanism Experimental limits on the nEDM lead to strong CP problem Peccei-Quinn (PQ) mechanism Θ → a(x): axion field R. D. Peccei and H. R. Quinn (1977) Θ vanishes dynamically

38. Axion potential where CP-violating interactions generate the linear term non-zero Θ is induced.

39. nEDM with PQ symmetry c.f.) M. Pospelov and A. Ritz (2001)

40. Higher-dimensional operators 4-Fermi operators Weinberg operator Their contribution is suppressed by light quark masses. comparable to the CEDM contribution generated by integrating out the 4-Fermi operators of heavy quarks J. Hisano, K. Tsumura, M. J. S. Yang, Phys. Lett. B713 (2012) 473.

41. Parameters for Condensates