1. The Weak Structure of the Nucleon from Muon Capture on 3He
D. Gazit, Phys. Lett. B 666, 472 (2008).
The Weak Structure of the Nucleon from Muon Capture on 3He
Doron Gazit
Institute for Nuclear Theory
University of Washington

2. The decay of a muonic 3He: competition
3He(m-,nm) p+2n
10%
3He(m-,nm) 3H
70%
3He
3He(m-,nm) d+n
20%
3He
p+2n
The rates become comparable for Z~10.
The Z4 law has deviations – mainly due to nuclear effects.
In order to probe the weak structure of the nucleon, one has to keep the nuclear effects under control.

3. In other words…
Solve the nuclear problem from the nucleonic degrees of freedom.
However:
This is possible only for light nuclei.
The nuclear effects inside the nucleus are complicated – in particular the weak interaction of the muon with meson currents in the nucleus.
However, a parameter free, percentage level accuracy calculation of the process is a great challenge to nuclear physics.
The MuCap collaboration (PSI) measuring:
For the (exclusive) process 3He(m-,nm) 3H an incredible measurement (0.3%):
Expecting to achieve 1% accuracy.
MuCap, Phys. Rev. Lett. 99, (2007).
Ackerbauer et al, Phys. Lett. B417, 224 (1998).

4. Muon weak interaction with the nucleus
Lepton current
Nuclear current
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5. Previous calculations…
Some calculations look at the nuclei as spin ½ doublet. However, too many free parameters.
Ab-initio calculations, based on phenomenological MEC or  excitation:
Congleton and Truhlik [PRC, 53, 956 (1996)]: 150232 Hz.
Marcucci et. al. [PRC, 66, (2002)]: 14844 Hz.
The main critique – too much freedom, without microscopic origin.
The modern point of view – chiral perturbation approach, with nucleons and pions as explicit degrees of freedom – an effective theory for low energy QCD.
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6. cPT approach for low-energy nuclear reactions:
Low energy EFT
QCD
The problem – for now:
Nuclear potentials were developed only recently, and only to third order.
Chiral Lagrangian
Nuclear Hamiltonian +
Solution of Schrödinger equation
Nöther current
Wave functions
Weak current
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7. EFT* approach for low-energy nuclear reactions:
Low energy EFT
QCD
Phenomenological Hamiltonian (with correct long-pion tail)
Chiral Lagrangian
Nöther current
Solution of Schrödinger equation
Wave functions
Weak current
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T.-S. Park et al, Phys. Rev. C 67, (2003), M. Rho arXiv: nucl-th/

8. The Nuclear Wave Functions
Barnea, Leidemann, Orlandini, PRC, (2001); Nucl. Phys. A, 693 (2001) 565.
The Nuclear Wave Functions
Previous calculations [Marcucci et. al] showed no significance sensitivity to the nuclear potential [±2 Hz], as long as the energies are reproduced.
Consistent cPT investigations show that the pion tail in the Hamiltonian is the important part, whereas short range correlations in the wave functions do not affect GT type of operators. [DG, Quaglioni, Navratil]
We use: AV18+UIX, and solve using EIHH approach
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9. EFT* approach for low-energy nuclear reactions:
Low energy EFT
QCD
Phenomenological Hamiltonian (with correct long-pion tail)
Chiral Lagrangian
Nöther current
Solution of Schrödinger equation
Wave functions
Weak current
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T.-S. Park et al, Phys. Rev. C 67, (2003), M. Rho arXiv: nucl-th/

10. cPT based weak currents to fourth order
Single nucleon current
Contact term
1 pion exchange
No free parameters in the meson exchange currents – We can investigate the nucleon.
Nucleon-pion interaction, NO free parameters
Contact term:
ONE free parameter.
Calibrated using triton b decay
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T.-S. Park et al, Phys. Rev. C 67, (2003); DG PhD thesis arXiv:

11. Single Nucleon Currentsp' q
Vector
Axial
Magnetic
Induced
Pseudo-Scalar
Second class currents p
Weinberg Phys. Rev., 112, 1375 (1958)

12. 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, we expect a non vanishing result -
Using QCD sum rules:
Shiomi, J. Kor. Phys. Soc., 29, S378 (1996)

13. Conserved Vector Current Hypothesis
The weak vector current is an isospin rotation of the electromagnetic current, and in particular conserved.
So, if CVC holds then:
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14. Result ?
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15. Radiative corrections to the process
Beta decay has prominent radiative corrections. Why not for muon capture?
Czarnecki, Marciano, Sirlin, showed that radiative corrections increase the cross section by 3.00.4%.
This worsens the good agreement of the old calculations.
But…
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Czarnecki, Marciano, Sirlin, Phys. Rev. Lett 99, (2007)

16. Final result:
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17. Constraints on the weak structure of the nucleon
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18. Induced pseudo-scalar:
From PT [Bernard, Kaiser, Meissner, PRD 50, 6899 (1994); Kaiser PRC 67, (2003)]:
From muon capture on proton [Czarnecki, Marciano, Sirlin, PRL 99, (2007); V. A. Andreev et. al., PRL 99, (2007)]:
This work:
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19. Induced Tensor: From QCD sum rules:
Experimentally [Wilkinson, Nucl. Instr. Phys. Res. A 455, 656 (2000)]:
This work:
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20. Induced scalar (limits CVC):
“Experimentally” [Severijns et. al., RMP 78, 991 (2006)]:
This work:
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21. Final remark
The current formalism correctly describes the weak process
A great success to the theory – correct structure of currents
The calculation is done without free parameters, thus - a prediction.
A fully consistent cPT calculation, when possible, could:
Decrease nuclear model dependence.
Increase reliability.
A more accurate estimate of radiative corrections is needed.
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