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1
Muon Capture by 3He and
The 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:
€
aBμ =
h
Zmμcα=
me
mμ
~1/ 207{
aBe
€
ψ1S r( ) =1
π
1
aBμ
⎛
⎝ ⎜
⎞
⎠ ⎟
3 / 2
e−r / aBμ
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.
€
μ−→ ν μ + e− + ν e
€
τ μfree = 2.197019(21) ×10−6 sec
€
τ μbound ∝
1
ψ 0( )2Z∝
1
Z 4
€
μ−+ZA X → ν μ +Z +1
A X
5
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.
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%):€
Γ μ−p → ν μ n( )1S
singlet= 725.0 ±13.7stat ±10.7syst Hz
€
μ−+3He →ν μ +t
€
Γ μ−+3He → ν μ +t( )stat
=1496 ± 4Hz
7
Kinematics
€
μ
( )iP,EP iì =i
( )ìP E ,Pf f f=
€
k1μ = mμ ,0( )
( )2k,kk 2ì2 =
€
q0μ = ω,q( )
€
ν μ
€
W -
€
3He
€
3H
€
q02 = −0.954 mμ
2
€
ψ1Sav 2
= R ψ1S (0)2
≈ 0.979ZαMR( )
3
π
€
ˆ H W = −GFVud
2d3 r
x ˆ j μ+∫
r x ( ) ˆ J μ − r
x ( )
Lepton current
Nuclear current
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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.
1
HA
i NN NNNi i j i j k
t v v= < < <
= + +∑ ∑ ∑
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.
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4-body system with4-body system with MT-V nucleon-MT-V nucleon-
nucleon potentialnucleon potential
EIHHBARE
Binding Energy
Matter Radius
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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:
€
rq =103.22 MeV/c q0 = 2.44 MeV
q2 = −0.954mμ2
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 --> HBPT.
€
ˆ J μ− =
τ −
2ˆ J μ
A + ˆ J μV
( )
Axial Vector
15
Single Nucleon Currents
€
ˆ J μV = u p'( ) FV q2( )γ
μ +i
2MN
FM q2( )σ
μν qν +gS
mμ
qμ ⎡
⎣ ⎢
⎤
⎦ ⎥u p( )
ˆ J μA = −u p'( ) GA q2( )γ
μγ 5 +GP q2
( )
mμ
γ 5qμ +
igt
2MN
σ μν γ 5qν
⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥u p( )
Vector Magnetic
Axial Induced Pseudo-Scalar
Second class currents
Weinberg PR, 112, 1375 (1958)
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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)]:
€
G = Ce iπT2
€
gS ~ gt ~mu − md
MN
€
gt
gA
= −0.0152(53)
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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:
€
gS = 0
limq →0
F1 q2( ) =1 lim
q →0FM q2
( ) = μ p − μn = 3.706...
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q Dependence of the Form Factors
€
at q2 = −0.954mμ2
FV q2( ) =1+
1
6r2 q2 = 0.974(1)
FM q2( ) = FM q2
( ) 1+1
6r2 q2 ⎛
⎝ ⎜
⎞
⎠ ⎟= 3.580(3)
GA q2( ) = gA 1+
1
6rA
2 q2 ⎛
⎝ ⎜
⎞
⎠ ⎟=1.245(4)
GP q2( ) =
2mμ gπpn fπ
mπ2 − qμ
2−
1
3gA mμ MN rA
2 = 7.99(20) Adler-Dothan Formula
19
HBPT systematics Identify Q – the energy scale of the process. (for μ
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).
2ν = +
nd
DerivativesDerivativesor pion massesor pion masses
nucleonsnucleons
order of interactionorder of interaction
20
Chiral Lagrangian (NLO)( )
( ) ( )
( ) ( )
5
2 2 2† †
1 5 2
Tr Tr U+U 24 4
4, Tr a
EFT A N
N
N i iv g a M N
f f mU U
i N a a N NN aM M
D N a NNN iD N iv NNN
μ μμ μ μ
μπ π πμ
μν μπ πμ ν μ
μ μμ μ μ
γ γ γ
κ βσ
γ γ γ
⎡ ⎤= ∂ + + − +⎣ ⎦
+ ∂ ∂ + − +
⎡ ⎤− + +⎣ ⎦
⎡ ⎤− + ∂ + +⎣ ⎦ K
L -N basic interaction
Lagrangian
N of order 3
2N contact terms
Calibrated using 3H life
time
21
MEC – back to configuration space
Fourier transform, with a cutoff .
{ } ( )( )
( ) ( )23
32π
ΛΛ
⋅= ∫ ik rf r k f kd k e S
r r rrr
Gaussian cutoff function
22
Resulting MEC
Park et. al. [PRC 67(2003), 055206]
€
rr =
r r 1 −
r r 2
S12 = 3r σ 1 ⋅ ˆ r
r σ 2 ⋅ ˆ r −
r σ 1 ⋅
r σ 2
Ok = τ + r σ k
T = ˆ r ̂ r ⋅O-1
3O
23
Remarks To one loop (relevant to N3LO), HBPT 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
€
Γ=2G2 Vud
2Eν
2
2J 3 He+1
1−Eν
M 3 H
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟ψ1s
av 2ΓN =1455 Hz
25
Previous results Ab-initio calculations, based on
phenomenological MEC or : Congleton and Truhlik [PRC, 53, 956
(1996)]: 150232 Hz. Marcucci et. al. [PRC, 66, 054003(2002)]:
14844 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 3.00.4%.
This ruins the good agreement of the old calculations.
But…
27
Final result:
€
Γ=2G2 Vud
2Eν
2
2J 3 He+1
1−Eν
M 3 H
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟ψ1s
av 2ΓN
⎧ ⎨ ⎪
⎩ ⎪
⎫ ⎬ ⎪
⎭ ⎪1+ RC( )
Γ =1499(2)Λ (3)NM (5)t (6)RC =1499 ±16 Hz
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.
€
μ−+3He →ν μ +t
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:
€
gP −0.954mμ2
( ) = 7.99(0.20)
€
gP −0.88mμ2
( ) = 7.3(1.2)
€
gP −0.954mμ2
( ) = 8.13(0.6)
30
Conclusions(iii) Induced Tensor:
From QCD sum rules: Experimentally [Wilkinson, Nucl. Instr. Phys. Res.A 455, 656
(2000)]:
This work:
€
gt
gA
= −0.0152(53)
€
gt
gA
< 0.36 at 90%
€
gt
gA
= −0.1(0.68)
31
Conclusions(iv) Induced Scalar: (limit on CVC)
“Experimentally” [Severijns et. al., RMP 78, 991 (2006)]:
This work: €
gS = 0.01± 0.27
€
gS = −0.005 ± 0.04
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 HBPT prediction? Theoretical methods for calculating weak form
factors: Lattice Holographic QCD (DG, Yee, in preparation)?
……