Transcript 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 ofWashington
The decay of a muonic 3He: competition
m
e
aBm 
e
Zmmc

~1/ 207
e

10%
3He


 m  2.197019(21) 106 sec
free
me e
aB
mm
3p+2n
He
70%
20%
3He(m,
m) p+2n
3He(m,
Capture prob. ~ Z  1S 0
 The rates become comparable for Z~10.
3He(m,
m) d+n
2
m)
3H
mm 3 4
~   Z
me 
 The Z4 law has deviations – mainly due to nuclear effects.
 In order to probe the
 weak structure of the nucleon, one has to keep
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the nuclear effects under control.
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.
The MuCap collaboration (PSI) measuring:
m p   m n
singlet
1S
 725.0  13.7 stat  10.7 syst Hz
Expecting to achieve 1% accuracy.
MuCap, Phys. Rev. Lett. 99, 032002 (2007).

For the (exclusive) process 3He(m,m) 3H
an incredible measurement ( 0.3%):
m  3 He   m +t 
stat
 1496  4 Hz
Ackerbauer et al, Phys. Lett. B417, 224 (1998).
Muon weak interaction with the nucleus
GV
Hˆ W   F ud
2


d 3 x ˆj m x Jˆ m

x 
m
k“2  k 2 , k 2 
m2
q20  0.954m
qm0  ,q
Nuclear 
current
Lepton
current
3


H
P “f  E f , P f

W-
k1m  mm ,0
Pi“  Ei , Pi 

ZM R 
 0.979
3
av 2
1S


 R 1S (0)
N 
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3
2

 3
ˆ
H Jm He
2


m
qm q
gm  2
MW
W propagator 
q 2  MW2
3
He

q M W
gm
MW2
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, 054003(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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cPT approach for low-energy nuclear reactions:
Low energy
EFT
QCD
The problem – for now:
Nuclear potentials
developed +
only recently,
Nuclearwere
Hamiltonian
and only to third order.
Solution of Schrödinger equation
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Nöther
current
Weak
current
Wave
functions
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Chiral Lagrangian
N 
3
H Jˆm 3He
2
EFT* approach for low-energy nuclear reactions:
Low energy
EFT
QCD
Phenomenological
Hamiltonian (with
Chiral Lagrangian
correct long-pion tail)
Nöther
current
Solution of Schrödinger equation
Weak
current
Wave
functions
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N 
3
H Jˆm 3He
2
T.-S. Park et al, Phys. Rev. C 67, 055206 (2003), M. Rho arXiv: nucl-th/061003.
Barnea, Leidemann, Orlandini, PRC, 63 057002
(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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EFT* approach for low-energy nuclear reactions:
Low energy
EFT
QCD
Phenomenological
Hamiltonian (with
Chiral Lagrangian
correct long-pion tail)
Nöther
current
Solution of Schrödinger equation
Weak
current
Wave
functions
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N 
3
H Jˆm 3He
2
T.-S. Park et al, Phys. Rev. C 67, 055206 (2003), M. Rho arXiv: nucl-th/061003.
cPT based weak currents to fourth order
Single nucleon current
1 pion exchange
Nucleon-pion interaction,
NO free parameters
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Contact term
Contact term:
ONE free parameter.
Calibrated using
triton b decay
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T.-S. Park et al, Phys. Rev. C 67, 055206 (2003); DG PhD thesis arXiv: 0807.0216
p'
Single Nucleon Currents
q
Jˆ
mV

i
gS m 
2
m
2
m
 u  p'FV q  
FM q  q 
q u p
2M N
mm 

Vector
Jˆ mA
Magnetic
Second class currents
2


g
q


ig
P
 u p'GA q 2  m  5 
 5q m  t  m  5q u p
mm
2M N




Axial

11Weinberg Phys. Rev., 112, 1375 (1958)
Induced
Pseudo-Scalar
p
Second class terms - G parity breaking
 G parity is the symmetry to a combined charge conjugation
and rotation in isospin space: G  Ce iT2
 Due to the fact that isospin is an approximate symmetry, we
expect a non vanishing result -
mu  md
gS 
~ gt ~
MN
 Using QCD sum rules:

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gt
 0.0152(53)
gA
Shiomi, J. Kor. Phys. Soc., 29, S378 (1996)
J mV 
gS m
q
mm
J mA 
igt
 m  5 q
2M N
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:
lim F1q 2  1 ; gS  0 ; lim FM q 2  m p  mn  3.706...
q 0
q 0
current
conservation
0
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
qmJ
mV
gS 2

q
mm
Result
2G Vud E2  E  av 2


1

1s N  1455 Hz


2J 3 He  1  M 3 H 
2
2
 exp  1496  4 Hz


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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, 032003 (2007)
Final result:
2
2

2G Vud E 2  E   av 2 

  

1

1s N 1 RC




 2J 3 He  1  M 3 H 

 =1499(2) (3) NM (5) t (6) RC  1499 16 Hz
EXP  1496  4 Hz
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Constraints on the weak
structure of the nucleon
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Induced pseudo-scalar:
 From cPT [Bernard, Kaiser, Meissner, PRD 50, 6899 (1994);
Kaiser PRC 67, 027002 (2003)]:
gP 0.954mm2  7.99(0.20)
 From muon capture on proton [Czarnecki, Marciano, Sirlin,
PRL 99, 032003 (2007); V. A. Andreev et. al., PRL 99,
032004(2007)]:

gP 0.88mm2  7.3(1.2)
 This work:

gP 0.954mm2  8.13(0.6)
2mm gpn f 1
2
gP q 

g
m
M
r
A m
N A  7.99(20)
2
2
m  qm
3
2
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
Induced Tensor:
 From QCD sum rules:
gt
 0.0152(53)
gA
 Experimentally [Wilkinson, Nucl. Instr. Phys. Res. A 455,
656 (2000)]:

gt
 0.36 at 90%
gA
 This work:


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gt
 0.1(0.68)
gA
J
mA
igt

 m  5 q
2M N
Induced scalar (limits CVC):
 “Experimentally” [Severijns et. al., RMP 78, 991 (2006)]:
gS  0.01  0.27
 This work:

gS  0.005  0.04

J
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mV
gS m

q
mm
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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