SU(2) chiral perturbation theory for Kl3 decay amplitudes

Abstract

We use one-loop $\SU(2)_L\times \SU(2)_R$ chiral perturbation theory ($\SU(2)$ ChPT) to study the behaviour of the form-factors for semileptonic $K\to \pi$ decays with the pion mass at $q^2=0$ and at $q_{\text{max}}^2=(m_K-m_{\pi }{)}^2$, where $q$ is the momentum transfer. At $q^2=0$, the final-state pion has an energy of approximately $m_K/2$ (for $m_K\gg m_{\pi }$) and so is not soft, nevertheless it is possible to compute the chiral logarithms, i.e. the corrections of $O(m_{\pi }^2\log (m_{\pi }^2))$. We envisage that our results at $q^2=0$ will be useful in extrapolating lattice QCD results to physical masses. A consequence of the Callan-Treiman relation is that in the $\SU(2)$ chiral limit ($m_u=m_d=0$), the scalar form factor $f^0$ at $\qsqmax$ is equal to $f^{(K)}/f$, the ratio of the kaon and pion leptonic decay constants in the chiral limit. Lattice results for the scalar form factor at $\qsqmax$ are obtained with excellent precision, but at the masses at which the simulations are performed the results are about 25% below $f^{(K)}/f$ and are increasing only very slowly. We investigate the chiral behaviour of $f^0(\qsqmax)$ and find large corrections which provide a semi-quantitative explanation of the difference between the lattice results and $f^{(K)}/f$. We stress the generality of the relation $f^0_{P\toπ}(\qsqmax)=f^{(P)}/f$ in the $\SU(2)$ chiral limit, where $P=K,D$ or $B$ and briefly comment on the potential value of using this theorem in obtaining physical results from lattice simulations.

20 pages, 4 figures

References in corpus (2)

• Kl3 semileptonic form factor from 2+1 flavour lattice QCD
• Full CKM matrix with lattice QCD

Cited by in corpus (2)

• $K\to \pi \pi$ Decays in SU(2) Chiral Perturbation Theory
• Next to Leading Order Chiral Perturbation theory of $K\pi \to \pi$ and $K\to \pi \pi$ amplitudes