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D 0 -D 0 Mixing

D 0 -D 0 Mixing. Department of Physics Nankai University. Mao-Zhi Yang ( 杨茂志 ). Jul. 23, 2008, KITPC. Program on Flavor Physics. Contents. I. Introduction. II. The basic formulas. III. The mixing parameters in the SM. IV Decays of physical neutral D system.

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D 0 -D 0 Mixing

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  1. D0 -D0 Mixing Department of Physics Nankai University Mao-Zhi Yang(杨茂志) Jul. 23, 2008, KITPC Program on Flavor Physics

  2. Contents I. Introduction II. The basic formulas III. The mixing parameters in the SM IV Decays of physical neutral D system V Time-dependent measurement of D0D0 mixing VI Summary

  3. and can transform into each under weak interaction and can not be separated absolutely I Introduction

  4. d s b u c t The D0-D0 mixing occurs via loop diagrams involving intermediate down-type quarks, it provides unique information about weak interaction In the standard model, themixing amplitude is quite small It is severely suppressed by the GIM mechanisms Loop-integration function The b-quark contribution is highly suppressed by the CKM factor

  5. The CKM suppression factor The b-quark contribution in the loop diagram can be neglected Thus, the mixing in D system involves only the first two generations. CP violation is absent in both the mixing and decay amplitudes, and therefore can be neglected. The mixing amplitude vanishes in the limit of SU(3) flavor symmetry , ms=md, due to the GIM suppression. Mixing is only the effect of SU(3) breaking

  6. II The basic formulas In general the neutral D meson exists as a mixture state of D0 and D0 Assume there is a neutral D state at t=0: Then at any time t, the state evolves into States D decays into Oscillation within neutral D state

  7. If we only consider the oscillation within the neutral D state, then we can consider the evolution of the following state which can be written in the form of matrix product then we can use to stand for the wave function of the neutral D meson state

  8. The Shrödinger equation for the evolution of the wave function is H needs not be Hermite because D meson can decay in the evolution

  9. The matrix H expressed explicitly in term of the matrix elements The matrix elements are determined by the Hamiltonians of strong, electromagnetic and weak interactions The magnitude of weak interaction is greatly smaller than the strong and electromagnetic interaction

  10. The eigenstates of Hst+Hem The weak interaction can be treated as a perturbation over the strong and EM interaction. Then the matrix elements can be solved perturbatively

  11. Using the formula Then from

  12. Theorems: ① If CPT is conserved, then M11=M22, and Γ11=Γ22. Γ12*M12* ② If T is conserved, then Γ12M12 The proof can be performed with the formulas in the previous page

  13. The eigen-equation Solve the equation, one can get the eigenvalues and eigenfunctions

  14. The imaginary parts will be decay widths of the two eigenstates: The real parts of will be the masses m1 and m2 of the two eigenstates That is

  15. The two relevant eigenstates are

  16. If CPT is conserved, then z=0 Then p and q satisfy the normalization condition

  17. III The estimation of the mixing parameters in the SM Two physical parameters that characterize the mixing are Where Γis the average decay widths of the two eigenstates D1 and D2

  18. Using the eigenvalues of D1 and D2, we can obtain with CPT conservation and neglecting CP violation in mixing where is implicitly included

  19. Define the correlator Insert a complete set of final states One can obtain Compare the above result with the expression of

  20. One can obtain

  21. The contribution of the box diagrams

  22. The box contribution to is given by

  23. Numerically, by simply taking which leads to The bare quark loop contribution to is even further suppressed by additional powers of Numerically, one finds

  24. The small result of box diagram can be enhanced by various long-distance effects, or by contributions of higher-dimension operator in the OPE Long-distance effects

  25. Long-distance contributions can severely enhance the mixing parameters, although it is difficult to calculate them accurately. It is estimated that long-distance dynamics can enhance the mixing parameters to be J.F. Donoghue et.al, Phys. Rev. D33, 179 (1986) E. Golowich, A.A. Petrov, Phys. Lett. B427, 172 (1998)

  26. Contribution of higher-dimension operator in operator product expansion (OPE) The time ordered product can be expanded in local operators of increasing dimension. The higher dimension operators are suppressed by powers of

  27. Higher order terms in the OPE can be important, because chiral suppression can be lifted by the quark condensates, which lead to contribution proportional to ms2, rather than ms4. Diagram for 6-quark operator (D=9) Diagram for 8-quark operator (D=12) I.I. Bigi, N.G. Uraltsev, Nucl. Phys. B592 (2001)92 A. Falk et al., Phys. Rev. D65 (2002) 054034

  28. For , the dominant contribution is from 8-quark operators Explicitly, the contribution of 6-quark operator with D=9 is given by The dominant contribution to is from 6- and 8-quark operators Numerically, the resulting estimates are

  29. This estimate means that the observation of mixing parameters severely larger than 10-3 would reveal the existence of new physics beyond the SM But this statement can not be conclusive because the uncertainty in this estimation is still large. The main uncertainty comes from the size of the relevant hadronic matrix elements. Larger observed values of x and y might indicate serious underestimation of the hadronic matrix elements.

  30. IV Decays of physical neutral D system How to measure the mixing parameters in experiments? To measure them, one has to study the evolution and various decays of the neutral D meson system.

  31. The evolution of neutral D meson state For any neutral D state at any time t, its wave function can be expanded as linear combination of the eigenstates constant c1 and c2 can be determined by initial and normalization condition

  32. For a state initially,i.e., Denote such a state as

  33. Then Similarly

  34. The time-dependent decay rate of a physical neutral D meson state The time-dependent amplitude of initial D0 physical state to any final state f

  35. ∴ A normalization factor

  36. We can get the decay rate of an initial D0 state by

  37. (1) Semi-leptonic decay At leading order of weak interaction ∴

  38. With mixing present, the wrong-sign decay can occur Let us consider the wrong-sign decays CP is violated!

  39. The condition for mixing induced CP violation CP violation in wrong-sign semileptonic decay We can get

  40. (2) Wrong-sign decay to hadronic final state f For Cabibbo favored Doubly Cabibbo suppressed

  41. Let us consider the doubly cabibbo suppressed decay Using the formua we have derived We can obtain

  42. define

  43. Similarly the CP conjugated process

  44. To simplify the above result, we can make the following definitions Mixing induced CPV In order to demonstrate the CP violation in decay, we define

  45. where

  46. CP violation in mixing decays with and without mixing CP violation in decay CP violation in the interference between 1. 2. 3.

  47. Then the time-dependent decay rates are Note the sign

  48. If CP is conserved, then Then

  49. The coefficients of the time-dependent terms can be fitted if the time dependent decay rate can be measured. CP: for CP:

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