The $K\to \pi$ vector form factor at zero momentum transfer on the lattice

Abstract

We present a quenched lattice study of the form factors $f_{+}(q^2)$ and $f_0(q^2)$ of the matrix elements $〈\pi |\overline{s}{\gamma }_{\mu }u|K〉$. We focus on the second-order SU(3)-breaking quantity $[1-f_{+}(0)]$, which is necessary to extract $|V_{us}|$ from $K_{\ell 3}$ decays. For this quantity we show that it is possible to reach the percent precision which is the required one for a significant determination of $|V_{us}|$. The leading quenched chiral logarithms are corrected for by using analytic calculations in quenched chiral perturbation theory. Our final result, $f_{+}^{K^0{\pi }^{-}}(0)=0.960\pm {0.005}_{\mathrm{stat}}\pm {0.007}_{\mathrm{syst}}$, where the systematic error does not include the residual quenched effects, is in good agreement with the estimate made by Leutwyler and Roos. A comparison with other non-lattice computations and the impact of our result on the extraction of $|V_{us}|$ are also presented.

Introduction

The most precise determination of the Cabibbo angle, or equivalently of the CKM matrix element $|V_{us}|$ [1], is obtained from $K\to \pi \ell \nu$ ($K_{\ell 3}$) decays. The key observation which allows to reach a good theoretical control on these transitions is the Ademollo–Gatto theorem [2], which states that the $K_{\ell 3}$ form factors, $f_{+}(q^2)$ and $f_0(q^2)$, at zero four-momentum transfer, are renormalized only by terms of at least second order in the breaking of the SU(3) flavor symmetry. The estimate of these smallish corrections, i.e., of the difference of $f_{+}(0)=f_0(0)$ from unity, is presently the dominant source of theoretical uncertainty in the extraction of $|V_{us}|$.

Chiral perturbation theory (CHPT) provides a natural and powerful tool to analyse the amount of SU(3) (and isospin) breaking due to light quark masses. As shown by Leutwyler and Roos [3], within CHPT one can perform a systematic expansion of the type $f_{+}(0)=1+f_2+f_4+\cdots$, where $f_n=\mathcal{O}[M_{K\text{,}\pi }^n/{\left(4\pi f_{\pi }\right)}^n]$. Because of the Ademollo–Gatto theorem, the first non-trivial term in the chiral expansion, $f_2$, does not receive contributions of local operators appearing in the effective theory and can be computed unambiguously in terms of $M_K$, $M_{\pi }$ and $f_{\pi }$ ($f_2=-0.023$ in the $K^0\to {\pi }^{-}$ case [3]). The problem of estimating $f_{+}(0)$ can thus be re-expressed as the problem of finding a prediction for

$$\mathrm{\Delta }f=1+f_2-f_{+}(0)\text{.}$$

This quantity is difficult to be evaluated since it depends on unknown coefficients of $\mathcal{O}(p^n)$ chiral operators, with $n⩾6$. Using a general parameterization of the SU(3) breaking structure of the pseudoscalar meson wave functions, Leutwyler and Roos estimated $\mathrm{\Delta }f=(0.016\pm 0.008)$. Very recently, Bijnens and Talavera [4] showed that, in principle, the leading contribution to Δf could be constrained by experimental data on the slope and curvature of $f_0(q^2)$; however, the required level of experimental precision is far from the presently available one. For the time being we are therefore left with the Leutwyler–Roos result, and the large scale dependence of the $\mathcal{O}(p^6)$ loop calculations [4], [5] seems to indicate that its 0.008 error might well be underestimated [6].

The theoretical error on $|V_{us}|$ due to the Leutwyler–Roos estimate of $f_{+}(0)$ is already comparable with the present experimental uncertainty (see, e.g., Ref. [7]). When the high-statistics $K_{\ell 3}$ results from KLOE and NA48 will be available, this theoretical error will become the dominant source of uncertainty on $|V_{us}|$. Given this situation, it is then highly desirable to obtain independent estimates of $f_{+}(0)$ at the $\approx 1\text{\%}$ level (or below). The purpose of the present work is to show that this precision can be achieved using lattice QCD.

The strategy adopted in order to reach the challenging goal of a $\approx 1\text{\%}$ error, is based on the following three main steps:

• Evaluation of the scalar form factor $f_0(q^2)$ at $q^2=q_{\max }^2={(M_K-M_{\pi })}^2$. Applying a method originally proposed in Ref. [8] to investigate heavy-light form factors, we extract $f_0(q_{\max }^2)$ from the relation

  $$\frac{〈\pi |\overline{s}{\gamma }_{0}u|K〉〈K|\overline{u}{\gamma }_{0}s|\pi 〉}{〈\pi |\overline{u}{\gamma }_{0}u|\pi 〉〈K|\overline{s}{\gamma }_{0}s|K〉}=\frac{{(M_K+M_{\pi })}^2}{4M_K{M}_{\pi }}{\left[f_0\left(q_{\max }^2\text{;}{M}_K\text{,}{M}_{\pi }\right)\right]}^2\text{,}$$

  where all mesons are at rest. The double ratio and the kinematical configuration allow to reduce most of the systematic uncertainties and to reach a statistical accuracy on $f_0(q_{\max }^2)$ well below 1%.

• Extrapolation of $f_0(q_{\max }^2)$ to $f_0(0)$. By evaluating the slope of the scalar form factor, we extrapolate $f_0$ from $q_{\max }^2$ to $q^2=0$. We note that in order to obtain $f_0(0)$ at the percent level the precision required for the slope can be much lower, because it is possible to choose values of $q_{\max }^2$ very close to $q^2=0$.

  For each set of quark masses we calculate two- and three-point correlation functions of mesons with various momenta in order to study the $q^2$ dependence of both $f_0(q^2)$ and $f_{+}(q^2)$. The latter turns out to be well determined on the lattice, whereas the former does not. We improve the precision in the extraction of $f_0(q^2)$ by constructing a new suitable double ratio which provides an accurate determination of the ratio $f_0(q^2)/f_{+}(q^2)$. We will define this ratio in Section 4. Fitting the $q^2$-dependence of $f_0(q^2)$ with different functional forms, we finally extrapolate $f_0(q_{\max }^2)$ to $f_0(0)$. The systematic error induced by this extrapolation, which is strongly reduced by the use of small values for $q_{\max }^2$, is estimated by the spread of the results obtained with different extrapolation functions.

• Subtraction of the leading chiral logs and chiral extrapolation. The Ademollo–Gatto theorem holds also within the quenched approximation [9], which has been adopted in this work. The leading $\mathcal{O}(p^4)$ chiral corrections to $f_0(0)$, denoted by $f_2^q$ where the superscript q refers to the quenched approximation, are finite and can be computed unambiguously in terms of the $\mathcal{O}(p^2)$ couplings of the quenched CHPT (qCHPT) Lagrangian [10], [11]. For these reasons, in order to get rid of some of the quenched artifacts we define the quantity

  $$R(M_K\text{,}{M}_{\pi })\equiv \frac{\mathrm{\Delta }f}{{\left(\mathrm{\Delta }{M}^2\right)}^2}=\frac{1+f_2^q(M_K\text{,}{M}_{\pi })-f_0(0\text{;}{M}_K\text{,}{M}_{\pi })}{{\left(\mathrm{\Delta }{M}^2\right)}^2}\text{,}$$

  where $\mathrm{\Delta }{M}^2\equiv M_K^2-M_{\pi }^2$ and extrapolate it to the physical kaon and pion masses. The ratio $R(M_K\text{,}{M}_{\pi })$: (i) is finite in the SU(3)-symmetric limit; (ii) does not depend on any subtraction scale; (iii) is free from the dominant quenched chiral logs. We emphasize that the subtraction of $f_2^q$ in Eq. (3) does not imply necessarily a good convergence of (q)CHPT at order $\mathcal{O}(p^4)$ for the meson masses used in our lattice simulations. The aim of this subtraction is to define the quantity Δf in such a way that its chiral expansion starts at order $\mathcal{O}(p^6)$ independently of the values of the meson masses. In the presence of sizable local contributions, we expect $R(M_K\text{,}{M}_{\pi })$ to have a smooth chiral behavior and to be closer to its unquenched analog than the SU(3)-breaking quantity $[1-f_{+}(0)]$. Extrapolating the values of $R(M_K\text{,}{M}_{\pi })$ to the physical meson masses, we finally obtain

  $$\mathrm{\Delta }f=R\left(M_K^{\mathrm{phys}}\text{,}{M}_{\pi }^{\mathrm{phys}}\right)\times {\left[{\left(\mathrm{\Delta }{M}^2\right)}^2\right]}^{\mathrm{phys}}=(0.017\pm {0.005}_{\mathrm{stat}}\pm {0.007}_{\mathrm{syst}})\text{,}$$

  where the systematic error does not include an estimate of quenched effects beyond $\mathcal{O}(p^4)$. Our result (4) is in good agreement with the estimate $\mathrm{\Delta }f=(0.016\pm 0.008)$ obtained by Leutwyler and Roos in Ref. [3].

  The systematic error quoted in Eq. (4) is mainly due to the uncertainties resulting from the functional dependence of the scalar form factor on both $q^2$ and the meson masses. This error can be further reduced by using larger lattice volumes (leading to smaller lattice momenta) as well as smaller meson masses. In our estimate of Δf discretization effects start at $\mathcal{O}(a^2)$ and are also proportional to ${(m_s-m_u)}^2$, as the physical SU(3)-breaking effects. In other words, our result is not affected by the whole discretization error on the three-point correlation function, but only by its smaller SU(3)-breaking part. Discretization errors on Δf are estimated to be few percent of the physical term, i.e., well within the systematic uncertainty quoted in Eq. (4). For a more refined estimate of these effects, calculations at different values of the lattice spacing are required. Finally, we stress again that the effects of quenching onto the terms of $\mathcal{O}(p^4)$ are not estimated and thus not included in our final systematic error.

Using the unquenched result for $f_2=-0.023$ (in the $K^0\to {\pi }^{-}$ case) [3], our final estimate for $f_{+}^{K^0{\pi }^{-}}(0)$ is given by

$$f_{+}^{K^0{\pi }^{-}}(0)=1+f_2-\mathrm{\Delta }f=0.960\pm {0.005}_{\mathrm{stat}}\pm {0.007}_{\mathrm{syst}}=0.960\pm 0.009\text{.}$$

The plan of the paper is as follows. In Section 2 we introduce the notation and give some details about the lattice simulation. Section 3 is devoted to the extraction of $f_0(q_{\max }^2)$ by means of the double ratio method, while in Section 4 we study the $q^2$ dependence of the form factors and extrapolate the scalar form factor to $q^2=0$. The calculation and the subtraction of the quenched chiral logarithms as well as the extrapolation of Δf to the physical masses is discussed in Section 5. The final estimate of $f_{+}(0)$, its comparison with non-lattice results and the impact on $|V_{us}|$ are discussed in Section 6. Finally our conclusions are given in Section 7.