Universit`a degli studi Roma Tre Measurement of the KL meson ...

16 CHAPTER 1. NEUTRAL KAON PHYSICS WITH KLOE
Figure 1.3: Scheme of the most common unitarity triangle.
1.5 Determination of the |Vus| matrix element
The |Vus| matrix element can be determined from hyperon, τ and kaon decays. Kaon
semileptonic decays, being 0 − → 0 − super-allowed Fermi transitions described only
by Vector Current matrix elements, offer the cleanest way to extract |Vus|.
The Vus matrix element appears in the kaon semileptonic decay width:
Γ(Kℓ3(γ)) = BR(Kℓ3(γ))
τK
=
= C2 KG2F M5 K
192π3 |Vus| 2 |f+(0)| 2 IKℓSEW
1 + δ SU(2)
K
+ δ EM
2 Kℓ
(1.33)
where K = K 0 , K ± , ℓ = e, µ and C 2 K=1/2, 1 for K ± and K 0 respectively. f+(0) ≡
fK0π −
+ (0) is the K → π transition form factor, the hadronic matrix element at zero
momentum transfer. Beside the value of |f+(0)|, theory provides SEW = 1.0232(3)
for the universal short-distance electroweak correction (ref. [26]), the SU(2)-breaking
δ SU(2)
K
(due to mu = md) and long-distance electromagnetic corrections δ EM
Kℓ , which
depend on the kaon charge and on the lepton flavor (see Ref. [27]), given in Tab. 1.1.
The dependence of the form factor on the momentum transfer, t = (P −p) 2 , where P
δ SU(2)
K (%) δ EM
Kℓ (%)
K 0 e3 0 +0.57(15)
K + e3 2.36(22) +0.08(15)
K 0 µ3 0 +0.80(15)
K + µ3 2.36(22) +0.05(15)
Table 1.1: Summary of the isospin-breaking corrections factors (see Ref. [28]). The
electromagnetic corrections factors refer to the fully-inclusive Kℓ3(γ) rate.
and p are respectively the kaon and the pion momentum, is measured from the pion
momentum spectrum; its knowledge is needed to evaluate the phase space integral
IKℓ. Beside this, experiments provide the measurement of the branching fraction,
BR(Kℓ3), the mass, MK, the lifetime, τK, and of the Fermi constant. This is the
best known quantity in Eq. 1.33: GF = 1.16637(1) × 10 −5 GeV −2 [18].