# Presentation: 847629

## Kaon semileptonic decay form factors from lattice QCD

- Naoto Tsutsui (KEK)
- for the JLQCD Collaboration
- PANIC05@Santa Fe
- October 25, 2005

## |Vus| from Kl3 decays

- high statistics experiments (E865, KLOE, KTeV, NA48) provide *Br(K**l3**)* with ≤1% accuracy
- dominant uncertainty comes from the theoretical estimate of the vector form factor *f**+**(0)*
- reliable calculation from the first principle of QCD is highly desirable

## f+(0) from chiral perturbation theory

*f**+**(0)=*1 in the SU(3) limit
- estimation of corrections from isospin breaking, SU(3) breaking is the key
- Leutwyler and Roos (1984)
*f**+**(0)=1+f**2**+f**4*=1-0.023-0.016(8)=0.961(8)
*f**2* is unambiguously determined by *f***and *m**PS* from ChPT
- rough estimate of *f**4* using a model of the wave function
- higher order and radiative correction
- ChPT at *O(p**6**) *and electromagnetic correction at *O(e**2**p**2**)*
- need better experimental resolution to determine LECs

## f+(0) from lattice QCD

- Becirevic et al.
- Nf=0, O(a)-improved Wilson + plaquette
*f**+**(0)=1+f**2**+f**4**q*=1-0.023-0.017(5)(7)=0.960(9)
- RBC collaboration
- Nf=2, domain-wall quark + DBW2
*f**+**(0)=*0.955(12)
- Fermilab/MILC/HPQCD collaboration
- Nf=2+1, improved staggerd + Symanzik
*f**+**(0)=*0.962(6)(9)

## JLQCD Nf=2 configurations

- gauge configurations include internal loop effects of up and down quarks
- Non-perturbatively O(a)-improved Wilson fermion + plaquette gauge
- 203x48, =5.2, a-1~0.1 fm
- 5 quark masses correspond to m=550~1000 MeV
- 1,200 configs separated by 10 HMC trajectories
- light hadron spectrum, decay constant and quark masses / *B* meson decay constant and Bag parameters

## 3-step strategy to calculate f+(0)

- determine *f**0* at* q**max**2**=(m**K**-m****)**2*
- interpolate to *q**2**=0*
- subtract unnecessary contribution from ***(0)=f**-**(0)/f**+**(0)*

-     To achieve a few percent accuracy,
-     a set of double ratio of correlation functions is used
- → renormalization factors and bulk of statistical fluctuations cancel

## Step 1: determine f0(qmax2)

- Double ratio used before for semileptonic *B* decay by the Fermilab group
- measures SU(3) breaking at *q**max**2**=(m**K**−**m****)**2*
- larger deviation from 1 for larger mass differences

- three-point function

- Ksea=0.1355 (lightest)

## Step 2: interpolate to q2=0

- 2pt. function in the denominator is to cancel the energy mismatch
- exactly 1 in zero-recoil case
- interpolate to *q**2**=0* with a quadratic function

- Ksea=0.1355 (lightest)

## Step 3: subtract  contribution

- q2 dependence is very small and seems to be independent of the strange quark mass
- extrapolation to q2=0 is done by assuming linear dependence

- Ksea=0.1355 (lightest)

## Chiral extrapolation

- the chiral logarithm is significant only in the region where m2 <<mK2, while the data region 1⁄2 < m2/mK2 < 2 is well described by the quadratic form
- partially quenched ChPT formula by Becirevic et al. is used

## Results for f+(0)

| Leutwyler and Roos (1984) | 0.961(8) |
| --- | --- |
| Becirevic et al. (2005) | 0.960(9) |
| This work (quad.) | 0.967(6) |
| This work (ChPT + quad.) | 0.952(6) |
| This work (pqChPT + quad.) | 0.945(6) |

## Vector charge radius

- charge radius is a slope of the form factor near *q**2**=0*
- one-loop ChPT predicts
- we fit the data with ChPT + linear

- cf) the exp. value

## Scalar charge radius

- The same analysis can be applied to the scalar charge radius
- overshoots the exp. value  0.21(3) fm      (PDG2004)

## Summary

- lattice QCD enables us to calculate the kaon form factor from the first principle of QCD
- a few percent statistical accuracy can be achieved by the double ratio method
- study of systematic errors should be made, especially, much lighter sea quarks are necessary for reliable chiral extrapolation
- details are presented in hep-lat/0510068