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

Muon Capture by 3He
and
The Weak Stucture of the Nucleon
Doron Gazit
Institute for Nuclear Theory
arXiv: 0803.0036
1
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.
2

Muon Capture


The muon binds to the atom.
Decays fast to the atomic ground state:
me e

aB 

aB
Zmc m
3/2


1 1
r / a B
1S r 
   e
 aB 
~1/ 207

3
Muon Capture - competing reactions

Free muon decay:
     e  e
 free  2.197019(21) 106 sec


Capture by the nucleus: 




 AZ X     AZ 1 X
Rate proportional to the overlap of the
nucleus size and the atomic wave function.

Rate proportional to the number of protons.
 bound 
1
 0 Z
2

1
Z4
4
Muon Capture - competing reactions



The Z4 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.
5
Muon Capture by a proton

MuCap Collaboration (PSI) on going
measurement:
 p   n
 725.0  13.7  10.7 Hz
singlet





syst
This 2.4% is expected to reach 1%.
For the (exclusive) process   He   +t
an incredible measurement (0.3%):
  He   +t   1496  4 Hz




stat
1S
3

3

stat
6
Kinematics
GV
Hˆ W   F ud
2


d 3 x ˆj  x Jˆ 
Lepton
current

x 

3
kμ2  k 2 , k 2 
2
q20  0.954m
q0  ,q
Nuclear
current


k1  m ,0
H
P μf   E f , P f
WPiμ  Ei , Pi 

ZM R 
 0.979
3
av 2
1S

 R 1S (0)
2




3
He
7

Multipole decomposition of the nuclear
current:

The entire nuclear contribution is:
8
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.
A
We use: H   ti   vNN 
i 1



i j
v
i  j k
NNN
AV18 - 2N potential
Urbana IX - 3N force
We use the effective interaction in the
hyperspherical harmonics method to solve the
problem.
9
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.
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Binding Energy
EIHH
4-body system with
MT-V nucleon-nucleon
potential
BARE
Matter Radius
11
Eexp=28.296 MeV
12
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:
q  103.22 MeV/c
q 0  2.44 MeV
q2  0.954m2
13
Weak Currents inside the Nucleus

The electro-weak theory dictates
only the structure of the currents:


Jˆ 
JˆA  JˆV
2

Axial

Vector
The muon can interact with:




A nucleon
(leading order).

Mesons inside the nucleus.
The currents reflect low energy
QCD --> HBPT.
14
Single Nucleon Currents
Jˆ
V

i
gS  
2

2

 u  p'FV q  
FM q  q 
q u p
2M N
m 

Vector
Jˆ A
Magnetic
Second class currents
2


G
q


ig
P
 u p'GA q 2    5 
 5q   t    5q u p
m
2M N




Axial
Induced
Pseudo-Scalar
 Weinberg PR, 112, 1375 (1958)
15
Second class terms - G parity breaking

G parity is the symmetry to a combined
charge conjugation and rotation in
isospin space: G  Ce iT
Due to the fact that isospin is an
approximate symmetry:
mu  md
2

gS ~ gt ~


MN
Using QCD sum rules: [Shiomi, J. Kor. Phys.
Soc., 29, S378 (1996)]: gt
 0.0152(53)
gA

16
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:
lim F1q 2  1
q 0
gS  0
lim FM q 2   p  n  3.706...
q 0
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q Dependence of the Form Factors
at q 2  0.954m2
FV q 2  1
1 2 2
r q  0.974(1)
6
1 2 2 
2
2 
FM q  FM q 1 r q  3.580(3)
 6

 1

GA q 2  gA 1 rA2 q 2  1.245(4)
 6

2m gpn f  1
2
2
GP q 

g
m
M
r
 7.99(20)
A

N
A
2
2
m   q
3
Adler-Dothan Formula
18
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).
n
 d
2
order of interaction
Derivatives
or pion masses
nucleons
19
Chiral Lagrangian (NLO)


LEFT  N i     iv   g A   5 a  M N N 
f2
f2 m2
 †

Tr    U  U  
Tr  U+U †  2  
4
4

4
i  N  a , a    N   NNTr  a  a   
MN
M


 D1 N    5 a NNN  iD2 N      iv  NNN 
-N basic interaction
Lagrangian
N of order 3
2N contact terms
Calibrated using
3H life time
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MEC – back to configuration
space

Fourier transform, with a cutoff L.
 f L  r   
d 3k
 2 
ik r
2
e
S
k

 f k 
L
3
Gaussian cutoff function
21
Resulting MEC
r  r1  r2
S12  31  rˆ2  rˆ  1  2
O k   k
Park et. al. [PRC 67(2003), 055206]
1
T  rˆrˆ  O - O
3
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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:
23
Results
2
2G 2 Vud E2  E  av 2


1

1s N  1455 Hz


2J 3 He  1  M 3 H 
24
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.
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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…
26
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)L (3) NM (5) t (6) RC  1499 16 Hz
27
Conclusions(i)




The current formalism correctly describes the
weak process   3 He    +t
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.
28
Conclusions(ii)

Induced Pseudo-scalar:



From PT [Bernard, Kaiser, Meissner, PRD 50, 6899 (1994);
Kaiser PRC 67, 027002 (2003)]:
gP 0.954m2  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)]:
2
g
0.88m

P
  7.3(1.2)
This work:

gP 0.954m2  8.13(0.6)
29
Conclusions(iii)

Induced Tensor:


gt
 0.0152(53)
gA
From QCD sum rules:
Experimentally [Wilkinson, Nucl. Instr. Phys. Res.A 455, 656
(2000)]:
g
 0.36 at 90%

g
t
A

This work:


gt
 0.1(0.68)
gA
30
Conclusions(iv)

Induced Scalar: (limit on CVC)

“Experimentally” [Severijns et. al., RMP 78,
991 (2006)]:
g  0.01  0.27
S

This work:

gS  0.005  0.04

31
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.
32
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)?
……
33