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$K^{+}topi^{+}pi^{0}$ decays at next-to-leading order in the chiral expansion on finite vol

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$K^{+}topi^{+}pi^{0}$ decays at next-to-leading order in the chiral expansion on finite vol

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$K^{+}topi^{+}pi^{0}$ decays at next-to-leading order in the chiral expansion on finite vol

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$K^{+}topi^{+}pi^{0}$ decays at next-to-leading order in the chiral expansion on finite vol

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$K^{+}topi^{+}pi^{0}$ decays at next-to-leading order in the chiral expansion on finite vol

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$K^{+}topi^{+}pi^{0}$ decays at next-to-leading order in the chiral expansion on finite vol

0209020v 1 3 S e p 20021

K +→π+π0decays at next-to-leading order in the chiral expansion on ?nite volumes ?

C.-J.

D.Lin a ?,G.Martinelli b ,

E.Pallante c ,C.T.Sachrajda a,d and G.Villadoro b

a

Dept.of Physics and Astronomy,Univ.of Southampton,Southampton SO171BJ,England b Dip.di Fisica,Universit′a di Roma “La Sapienza”,Piazzale A.Moro 2,I-00185Roma,Italy c SISSA,Via Beirut 2-4,34013,Trieste,Italy d Theory Division,CERN,CH-1211Geneva 23,Switzerland We present the ingredients for determining K +→π+π0matrix elements via the combination of lattice QCD and chiral perturbation theory (χPT).By simulating these matrix elements at unphysical kinematics,it is possible to determine all the low-energy constants (LECs)for constructing the physical K +→π+π0amplitudes at next-to-leading order (NLO)in the chiral expansion.In this work,the one-loop chiral corrections are calculated for arbitrary meson four-momenta,in both χPT and quenched χPT (q χPT),and the ?nite-volume e?ects are studied.1.INTRODUCTION The need for a high-precision prediction for K →ππamplitudes is underlined by the re-cent experimental measurement of Re(?′/?)and the long-standing puzzle,the ?I =1/2rule.Al-though the ?nite-volume (FV)techniques devel-oped in Refs.[3,4,5]can ultimately enable an accurate calculation of K →ππdecay rates,the most practical approach to the numerical calcu-lation of these decay rates remains the combina-tion of lattice QCD and (quenched and partially quenched)χPT.Apart from a calculation for the

CP-conserving,?I =3/2,K →ππdecay in

Ref.[1],all the numerical studies hitherto fol-

low a strategy proposed in Ref.[2],which only

allows the determination of these amplitudes at

leading-order (LO)in the chiral expansion 3.Be-

cause of the large kaon mass and the presence of

?nal state interactions,non-LO corrections in this

expansion are signi?cant.In a recent work [6,7],

we have proposed to perform lattice simulations

at unphysical kinematics over a range of meson

masses and momenta,from which we can deter-2(ˉs αd α)L q =u,d,s e q (ˉq βq β)R ,(1)

Q 8=34Another choice is considered in Ref.[8].

5Such a calculation for π+π0|Q 4|K + at two particular

kinematics,M K =M πand M K =2M πwith all mesons

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