2. 1 Muon Capture by 3 He and The Weak Stucture of the Nucleon Doron Gazit Institute for Nuclear Theory arXiv: 0803.0036

3. 2 Introduction The capture of negative muons by nuclei has been studied for 50 years. Played major role in the development of the weak interaction physics. Used to study nuclear structure, and its interplay with the weak force. Today - test QCD, BSM. Precision experiment and theory needed.

4. 3 Muon Capture The muon binds to the atom. Decays fast to the atomic ground state:

5. 4 Muon Capture - competing reactions Free muon decay: Capture by the nucleus: Rate proportional to the overlap of the nucleus size and the atomic wave function. Rate proportional to the number of protons.

6. 5 Muon Capture - competing reactions The Z 4 law has deviations, mainly due to processes inside the nucleus. The decay rates become comparable for Z~10. As a result: Less than one percent in hydrogen is due to capture. Hard to measure for protons or light nuclei, where the theory is clean.

7. 6 Muon Capture by a proton MuCap Collaboration (PSI) on going measurement: This  2.4% is expected to reach  1%. For the (exclusive) process an incredible measurement (  0.3%):

8. 7 Kinematics Lepton current Nuclear current

9. 8 Multipole decomposition of the nuclear current: The entire nuclear contribution is:

10. 9 Solving the Nuclear Problem Marcucci et. al. [PRC 66, 054003(2002)] : The capture rate depends weakly (  2 Hz) on the nuclear potential as long as the binding energies are reproduced. We use: AV18 - 2N potential Urbana IX - 3N force We use the effective interaction in the hyperspherical harmonics method to solve the problem.

11. 10 Effective Interaction in the Hyperspherical Harmonics method The HH - eigenfunctions of the kinetic energy operator, with quantum number K. We expand the WF in (anti) symmetrized HH. Use Lee-Suzuki transformation to replace the bare potential with an effective one. Barnea, Leidemann, Orlandini, PRC, 63 057002 (2001); Nucl. Phys. A, 693 (2001) 565.

12. 11 4-body system with MT-V nucleon-nucleon potential MT-V nucleon-nucleon potential EIHH BARE Binding Energy Matter Radius

13. 12 E exp =28.296 MeV

14. 13 The Nuclear Wave functions 3 He and 3 H are J  =1/2 + nuclei. Thus, the contributing multipoles can have J=0 or J=1, only. The resulting kinematics:

15. 14 Weak Currents inside the Nucleus The electro-weak theory dictates only the structure of the currents: The muon can interact with: A nucleon (leading order). Mesons inside the nucleus. The currents reflect low energy QCD --> HB  PT. Axial Vector

16. 15 Single Nucleon Currents Vector Magnetic AxialInduced Pseudo-Scalar Second class currents Weinberg PR, 112, 1375 (1958)

17. 16 Second class terms - G parity breaking G parity is the symmetry to a combined charge conjugation and rotation in isospin space: Due to the fact that isospin is an approximate symmetry: Using QCD sum rules: [ Shiomi, J. Kor. Phys. Soc., 29, S378 (1996) ]:

18. 17 Conserved Vector Current Hypothesis The weak vector current is an isospin rotation of the electromagnetic current, and in particular conserved. Thus, relations between multipoles. So, if CVC holds then:

19. 18 q Dependence of the Form Factors Adler-Dothan Formula

20. 19 HB  PT systematics Identify Q – the energy scale of the process. (for  capture– 100 MeV) In view of Q -Identify the relevant degrees of freedom. (pions and nucleons).  Choose  – the theory cutoff. (400-800 MeV) Write all the possible operators which agree with the symmetries of the underlying theory (QCD). Derivatives or pion masses nucleons order of interaction

21. 20 Chiral Lagrangian (NLO)  -N basic interaction  Lagrangian  N of order 3 2N contact terms Calibrated using 3 H life time

22. 21 MEC – back to configuration space Fourier transform, with a cutoff . Gaussian cutoff function

23. 22 Resulting MEC Park et. al. [PRC 67(2003), 055206]

24. 23 Remarks To one loop (relevant to N 3 LO), HB  PT gives the same single nucleon form factors. This is EFT* of Park et. al. [ PRC 67(2003), 055206 ], and the operators are the same. The operators shown - the numerically important [ Song et. al. PLB 656, 174 (2007) ] We are left with one unknown parameter: d r, which reflects a contact interaction. This parameter is calibrated using the experimental triton half life. Using a new measurement of the triton half life [ Akulov and Mamyrin, PLB 610, 45(2002) ] gives:

25. 24 Results

26. 25 Previous results Ab-initio calculations, based on phenomenological MEC or  : Congleton and Truhlik [PRC, 53, 956 (1996)]: 1502  32 Hz. Marcucci et. al. [PRC, 66, 054003(2002)]: 1484  4 Hz.

27. 26 Radiative corrections to the process Beta decay has prominent radiative corrections. Why not for muon capture? Recently,Czarnecki, Marciano, Sirlin PRL 99, 032003 (2007), showed that radiative corrections increase the cross section by 3.0  0.4%. This ruins the good agreement of the old calculations. But…

28. 27 Final result:

29. 28 Conclusions(i) The current formalism correctly describes the weak process The calculation is done without free parameters, thus can be considered as a prediction. One can do the reverse process and calibrate the unknown form factors (G P, g s, g t ). This constrain is the experimental constrain on the form-factors, from this reaction.

30. 29 Conclusions(ii) Induced Pseudo-scalar: From  PT [ Bernard, Kaiser, Meissner, PRD 50, 6899 (1994); Kaiser PRC 67, 027002 (2003) ]: From muon capture on proton [ Czarnecki, Marciano, Sirlin, PRL 99, 032003 (2007); V. A. Andreev et. al., PRL 99, 032004(2007) ]: This work:

31. 30 Conclusions(iii) Induced Tensor: From QCD sum rules: Experimentally [ Wilkinson, Nucl. Instr. Phys. Res.A 455, 656 (2000) ]: This work:

32. 31 Conclusions(iv) Induced Scalar: (limit on CVC) “Experimentally” [Severijns et. al., RMP 78, 991 (2006)]: This work:

33. 32 Conclusion of the Conclusions The use of muon capture on 3 He provides important and new limits to the induced pseudo-scalar, second class axial term and CVC term! One can increase the accuracy by reevaluating the triton half-life and by improving the radiative corrections calculations.

34. 33 Q Can the calculation be regarded as experimental extraction of the form factors? What is the difference between this calculation and older ones? Is this HB  PT prediction? Theoretical methods for calculating weak form factors: Lattice Holographic QCD (DG, Yee, in preparation)? ……