Muon Capture by 3 He and The Weak Stucture of the Nucleon

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

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.

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

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.

5. Muon Capture - competing reactions • The Z4law 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.

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

7. Kinematics Lepton current Nuclear current

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

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.

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.

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

12. Eexp=28.296 MeV

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

14. Vector Axial 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.

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

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)]:

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:

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

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 L – the theory cutoff. (400-800 MeV) • Write all the possible operators which agree with the symmetries of the underlying theory (QCD). nucleons order of interaction Derivatives or pion masses

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

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

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

23. Remarks • To one loop (relevant to N3LO), 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: dr, 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:

24. Results

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.

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…

27. Final result:

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 (GP, gs, gt). • This constrain is the experimental constrain on the form-factors, from this reaction.

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:

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

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

32. Conclusion of the Conclusions • The use of muon capture on 3He 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.

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)? • ……