Normal polarization component (PN)

of the K decay for the form factor determination

Dang Nguyen PhuongDepartment of Nuclear Physics, HCMUNS

The E06 meeting, Sep 1-2, 2007

Form factor of K3

K3 matrix element:

M ~ f+(q2) (pK

+ p) + f–(q

2) (pK – p

)

Linear expansion:

f±(q2) = f±(0) [1 + ± (q

2/m2)]

Quadratic expansion:

f±(q2) = f±(0) [1 + ± (q

2/m2) + ½ ± (q

4/m4) ]

q = pK - p

~ f+(q2) [ 2pK

. ū (1 – 5) v + ((q2) – 1)mū(1 – 5)v

(q2) = f–(q2) / f+(q2)

Form factor determination from K3 decay

Dalitz plot:

K+→

€

ρ(Eπ ,Eμ )∝ f+2 (q2) A + Bξ (q2) + Cξ 2 (q2)[ ]

€

A = mK (2Eμ Eν − mK ′ E π ) + mμ2 1

4′ E π − Eν

⎛

⎝ ⎜

⎞

⎠ ⎟

€

B = mμ2 Eν −

1

2′ E π

⎛

⎝ ⎜

⎞

⎠ ⎟

€

C =1

4mμ

2 ′ E π

€

E π = Eπmax − Eπ = (mK

2 + mπ2 + mμ

2 ) / 2mK − Eπ

Branching ratio:

K3 polarization:

Form factor determination from K3 decay

K+→

€

Γ(Kμ 3) /Γ(Ke3) = A + Bλ + + Cξ (0) + Eξ 2 (0) + Fλ +ξ (0)

€

rA = a1(ξ )

r p μ − a2 (ξ )

r p μmμ

mK − Eπ +

r p π ⋅

r p μ

r p μ

2(Eμ − mμ )

⎛

⎝

⎜ ⎜

⎞

⎠

⎟ ⎟+

r p π

⎡

⎣

⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥+ mK Imξ (t)(

r p π ×

r p μ )

€

PL =

r σ μ ⋅

r p μ

r p μ

( ))(

)(

μπμ

μπμμσ

ppp

pppPN rrr

rrrr

××

××⋅=

€

PT =

r σ μ ⋅(

r p π ×

r p μ )

r p π ×

r p μ

AArrr

/=σ

Results of

Branching ratio:

K3 polarization:

Dalitz plot: - 0.116 ± 0.030 (YUSHCHENKO)

- 0.13 ± 0.006 (KL3FIT)

- 0.95 ± 0.03 (CUTTS)- 1.0 ± 0.03 (BETTELS)

- 0.27 ± 0.25 (WHITMAN)

- 0.57 ± 0.24 (MERLAN)

Normal polarization component (PN)

( ))(

)(

μπμ

μπμμσ

ppp

pppPN rrr

rrrr

××

××⋅=

K+

PN

K+

bar

e+