1. Theoretical Calculation for neutron EDM Natsumi Nagata Based on J. Hisano, J. Y. Lee, N. Nagata, Y. Shimizu, Phys. Rev. D 85 (2012) 114044, K. Fuyuto, J. Hisano, N. Nagata, Phys. Rev. D 87 (2013) 054018. 13 July, 2014 NHWG10 University of Toyama Kavli IPMU → Univ. of Minnesota

2. 1.Introduction 2.Neutron EDM with QCD Sum Rules 3.Neutron EDM with Chiral Perturbation Theory 4.Conclusion & Prospects Outline

3. 1. Introduction

4. Motivation from Higgs physics Higgs (scalar) sector in general provides new sources for CP-violation. Ex.) 2HDM (with Z 2 symmetry) Additional phases induce CP-odd flavor-conserving quantities.

5. Electric Dipole Moments (EDMs) Hamiltonian E: Electric field s f : spin CP (CPT theorem) P- and T-odd quantity

6. Current limits Electron EDM ThO molecule ACME Collaboration, 1310.7534. Neutron EDM Institut Laue-Langevin, Phys.Rev.Lett. 97, 131801 (2006). Various proposal for future experiments Sensitivities are expected to be improved by about two orders of magnitude.

7. EDMs from CKM matrix in the Standard Model CKM matrix A source of CP-violation in the SM EDMs provide clean, background-free probes for CP-violating interactions in new physics. Its contribution to EDMs is quite suppressed because it is accompanied by flavor-changing interactions. Electron EDM Neutron EDM I. B. Khriplovich and A. R. Zhitnitsky (1982), T. Mannel and N. Uraltsev (2012). M. Pospelov and A. Ritz (2013).

8. Implication for new physics Ex.) 2HDM (with Z 2 symmetry) Electron EDM Neutron EDM T. Abe, J. Hisano, T. Kitahara, K. Tobioka [1311.4704]. It is important to consider constraints from various EDMs.

9. Neutron EDM New physics contribution to CP-odd quantities are expressed in terms of the parton-level interactions. Goal  Large uncertainty coming from non-perturbative hadronic effects.  Different methods give different results. Experimental limit on nEDM Experimental limit on nEDM Limits on CP @ parton-level Limits on CP @ parton-level 1.QCD Sum Rules 2.Chiral Perturbation Theory The difference itself should be regarded as “theoretical error”.

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

11. quark EDMsquark CEDMs θ term (QCD scale; up to dimension 5) Effective Lagrangian Suppressed by PQ symmetry The effects of new physics are expressed in terms of the operators. Remark The operators are induced at a scale much higher than the hadron scale. RGE effects are important. G. Degrassi, E. Franco, S. Marchetti, L. Silvestrini [hep-ph/0510137]. J. Hisano, K. Tsumura, M. J. Yang [arXiv:1205.2212]. Heavy quark effects

12. 2. Neutron EDM with QCD Sum Rules

13. 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) Correlation function of neutron interpolating fields Dispersion relation + div. terms (q 2 → - ∞) In the case of q 2 << 0, one can calculate it perturbatively (quark picture) Im Π(s): spectrum function It encodes information of Hadron spectrum Borel transformation eliminates div. terms and suppresses the contribution from excited resonances and continuum states.

14. Neutron interpolating field Neutron interpolating field: η n (x) A composite operator of quark fields with the same quantum numbers as those of neutron. An appropriate linear combination is chosen to suppress the higher order effects. Wave function Y. Aoki et. al. (2008) We extract the value of λ n from the lattice simulations.

15. Π(q 2 ) (q 2 << 0) is evaluated in terms of Short-distance contribution included in the Wilson coefficients C i (Q 2 ) Long-distance contribution included in the VEVs (condensates) Operator product expansion (OPE) Left-hand side … We carried out the NLO calculation. the operator product expansion (OPE).

16. OPE result Left-hand side 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 elsewhere. PQ symmetry is assumed.

17. Hadron contribution Right-hand side N N N N ＊ N ＊ N ＊ Contribution from excited states and continuum states. The effects of excited/continuum states are taken into account as theoretical error.

18. QCD sum rule for neutron EDM Now let us equate the above results and execute the Borel transformation. QCD sum rule Results (excited/continuum states) (OPE) (Lattice) Numerical results J. Hisano, J. Y. Lee, N.Nagata, Y. Shimizu (2012). c.f.) Previous result M. Pospelov and A. Ritz (2001)

19. Discussion  About factor-2 uncertainty  Relatively small value for neutron EDM (consistent with previous result, though).  No strange contribution The value of λ n from the lattice simulations is larger than those evaluated with QCD Sum rules. QCD Sum rules cannot include sea-quark effects appropriately ??

20. 3. Chiral perturbation theory

21. Chiral loop calculation CP-odd baryon-meson couplings The effects of CP-violation are included in the couplings. Loop calculation Λ: UV cut-off of the order of baryon mass Theoretical uncertainty

22. CP-odd baryon-meson couplings PCAC relations J 5 A : axial current Results Nucleon matrix elements We use lattice values. QCD Sum rules Remark Strange quark content is much smaller than those evaluated from baryon mass splitting.

23. Result We express the result as Since usually the quark CEDMs are proportional to the quark masses. Although there exists large uncertainty, the strange quark contribution tends to remain important. K. Fuyuto, J. Hisano, N.Nagata (2013).

24. Discussion  Large uncertainty  Larger value for neutron EDM  Sizable strange contribution UV cut-off, contribution from higher-dimensional operators... though the strange content in nucleon is small. Strange CEDM contribution remains sizable even when S s = D s = 0. Strange CEDM may be important.

25. 4. Conclusion & Prospects

26. Discussion  Precise calculation of neutron EDM is important to understand the phase structure of Higgs potential.  QCD sum rule method offers a systematic way of evaluating the neutron EDM, though it seems to lack the sea quark contribution.  Calculation with chiral perturbation theory has large uncertainty, but it can include the sea quark effects.  Pay attention when you use the results. Take care with possible unknown error in each method.

27. Appendix

28. 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

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

30. 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

31. 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

32. 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

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

34. 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

35. 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)

36. 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.

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

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

39. 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.

40. 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.

41. 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

42. 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.

43. 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 CEDMs and 4-Fermi operators of heavy quarks J. Hisano, K. Tsumura, M. J. S. Yang, Phys. Lett. B713 (2012) 473.