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Muon Capture by 3He and

The Weak Stucture of the Nucleon

Doron Gazit

Institute for Nuclear Theory

arXiv: 0803.0036

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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.

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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

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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

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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.

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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

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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( )

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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:

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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.

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HA

i NN NNNi i j i j k

t v v= < < <

= + +∑ ∑ ∑

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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

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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

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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

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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

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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

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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

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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

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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

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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:

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Results

€

Γ=2G2 Vud

2Eν

2

2J 3 He+1

1−Eν

M 3 H

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟ψ1s

av 2ΓN =1455 Hz

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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.

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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 3.00.4%.

This ruins the good agreement of the old calculations.

But…

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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

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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

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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)

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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)

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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

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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.

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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)?

……