Semileptonic Kaon Decay in Staggered Chiral Perturbation Theory

The determination of $\vert V_{us}\vert$ from kaon semileptonic decays requires the value of the form factor $f_+(q^2=0)$, which can be calculated precisely on the lattice. We provide the one-loop partially quenched staggered chiral perturbation theory expressions that may be employed to analyze staggered simulations of $f_+(q^2)$ with three light flavors. We consider both the case of a mixed action, where the valence and sea sectors have different staggered actions, and the standard case where these actions are the same. The momentum transfer $q^2$ of the form factor is allowed to have an arbitrary value. We give results for the generic situation where the $u$, $d$, and $s$ quark masses are all different, $N_f=1+1+1$, and for the isospin limit, $N_f=2+1$. The expression we obtain for $f_+(q^2)$ is independent of the mass of the (valence) spectator quark. In the limit of vanishing lattice spacing, our results reduce to the one-loop continuum partially quenched expression for $f_+(q^2)$, which has not previously been reported in the literature for the $N_f=1+1+1$ case. Our expressions have already been used in staggered lattice analyses of $f_+(0)$, and should prove useful in future calculations as well.Comment: 33 pages, 5 figures; v2: some referencing change

Error estimates for pi-pi scattering threshold parameters in Chiral Perturbation Theory to two loops

Using the analysis of ChPT to two loops, we perform an error analysis of the threshold and low energy parameters, based on the uncertainties for the one loop low energy parameters and the resonance saturation mechanism. Different sets of one loop low energy constants have been considered.Thus, the predictive power of the effective field theory is quantified on the basis of the present experimental uncertainties.Comment: 12 pages, 2 tables, 3 figures. Numerics upgraded to v2 of ref.(8). New reference added. Typos corrected. New figure for SU(3) parameters include

Two-Point Functions and S-Parameter in QCD-like Theories

We calculated the vector, axial-vector, scalar and pseudo-scalar two-point functions up to two-loop level in the low-energy effective field theory for three different QCD-like theories. In addition we also calculated the pseudo-scalar decay constant $G_M$. The QCD-like theories we used are those with fermions in a complex, real or pseudo-real representation with in general n flavours. These case correspond to global symmetry breaking pattern of $SU(n)_L\times SU(n)_R\to SU(n)_V$, $SU(2n)\to SO(2n)$ or $SU(2n)\to Sp(2n)$. We also estimated the S parameter for those different theories.Comment: 29 page

Strong Isospin-Breaking Effects in K-->pi pi at Next-to-Leading Order in the Chiral Expansion

Strong isospin-breaking (IB) contributions to both the octet and 27-plet weak K-->2 pi transitions are evaluated at next-to-leading order (NLO) in the chiral expansion. NLO contributions are shown to significantly reduce the leading order result for the potentially large contribution to the $\Delta I=3/2$ amplitude resulting from strong isospin-breaking modifications to the weak Delta I=1/2 amplitude. The ratio of strong IB 27-plet to strong IB octet contributions is found to be small for all decay amplitudes. Combined with recent results on the corresponding electromagnetic contributions, we find that the ratio of the intrinsic strengths of octet and 27-plet effective weak operators can be taken to be that obtained from experimental data, analyzed ignoring isospin breaking, to an accuracy better than of order 10 percent.Comment: 10 pages, 1 figure. NLO determination of strong IB modification of weak 27-plet transitions now also included; modified phase discussio

Quark-Resonance model

We construct an effective Lagrangian for low energy hadronic interactions through an infinite expansion in inverse powers of the low energy cutoff $\Lambda_\chi$ of all possible chiral invariant non-renormalizable interactions between quarks and mesons degrees of freedom. We restrict our analysis to the leading terms in the $1/N_c$ expansion. The effective expansion is in (\mu^2/\cutoff^2 )^P \ln (\cutoff^2/\mu^2 )^Q. Concerning the next-to-leading order, we show that, while the pure \mu^2/\cutoff^2 corrections cannot be traced back to a finite number of non renormalizable interactions, those of order (\mu^2/\cutoff^2 ) \ln (\cutoff^2/\mu^2 ) receive contributions from a finite set of 1/\cutoff^2 terms. Their presence modifies the behaviour of observable quantities in the intermediate $Q^2$ region. We explicitely discuss their relevance for the two point vector currents Green's function.Comment: 41 pages, 11 figures, preprint ROM2F 93/3

An upper bound on the Kaon B-parameter and Re(epsilon_K)

New precise data in B physics and theoretical developments in K physics lead us to reconsider the weak K^0-\bar{K}^0 transition from a large-N_c viewpoint, N_c being the number of colors. In this framework, we infer an upper limit on \hat{B}_K and the Kaon indirect CP violation.Comment: 11 pages, 4 figures. V2 : Minor corrections, final version accepted for publication in JHE

Order p^6 chiral couplings from the scalar K Pi form factor

Employing results from a recent determination of the scalar KPi form factor F_0^KPi within a coupled channel dispersion relation analysis \cite{JOP01}, in this work we calculate the slope and curvature of F_0^KPi(t) at zero momentum transfer. Knowledge of the slope and curvature of the scalar KPi form factor, together with a recently calculated expression for F_0^KPi(t) in chiral perturbation theory at order p^6, enable to estimate the O(p^6) chiral constants C_12^r=(0.3 +- 5.4)10^-7 and (C_12^r+C_34^r)=(3.2 +- 1.5)10^-6. Our findings also allow to estimate the contribution coming from the C_i to the vector form factor F_+^KPi(0) which is crucial for a precise determination of |V_us| from K_l3 decays. Our result F_+^KPi(0)|_C_i^r=-0.018 +- 0.009, though inflicted with large uncertainties, is in perfect agreement with a previous estimate by Leutwyler and Roos already made twenty years ago.Comment: 19 pages, discussion of scale dependence of the chiral couplings added; version to appear in JHE

Two-point function of strangeness-carrying vector-currents in two-loop Chiral Perturbation Theory

We calculate the correlator between two external vector-currents having the quantum-numbers of a charged kaon. We give the renormalized expression to two loops in standard chiral perturbation theory in the isospin limit, which, as a physical result, is finite and scale-independent. Applications include a low energy theorem, valid at two loop order, of a flavor breaking combination of vector current correlators as well as a determination of the phenomenologically relevant finite $O(p^6)$-counterterm combination $Q_V$ by means of inverse moment finite energy sum rules. This determination is less sensitive to isospin-breaking effects than previous attempts.Comment: 24 pages, revtex, 4 figures, 2 tables, revised version, one ref. adde

Charged Kaon K \to 3 pi CP Violating Asymmetries at NLO in CHPT

We give the first full next-to-leading order analytical results in Chiral Perturbation Theory for the charged Kaon K \to 3 pi slope g and decay rates CP-violating asymmetries. We have included the dominant Final State Interactions at NLO analytically and discussed the importance of the unknown counterterms. We find that the uncertainty due to them is reasonable just for \Delta g_C, i.e. the asymmetry in the K^+ \to pi^+ pi^+ pi^- slope g; we get \Delta g_C = -(2.4 +- 1.2) 10^{-5}. The rest of the asymmetries are very sensitive to the unknown counterterms. In particular, the decay rate asymmetries can change even sign. One can use this large sentivity to get valuable information on those counterterms and on Im(G_8) coupling --very important for the CP-violating parameter epsilon'_K-- from the eventual measurement of these asymmetries. We also provide the one-loop O(e^2 p^2) electroweak octet contributions for the neutral and charged Kaon K \to 3 pi decays.Comment: 43+2 pages, 2 figures. Version accepted in JHEP. Small changes in the final numerics of CP asymmetries due to change in input valu

Relations at Order p6p^6 in Chiral Perturbation Theory

We report on a search of relations valid at order $p^6$ in Chiral Perturbation Theory. We have found relations between $\pi\pi$, $\pi K$ scattering, $K_{\ell4}$ decays, masses and decay constants and scalar and vector form factors. In this paper we give the relations and a first numerical check of them.Comment: 18 pages, numerical discussion extended including a new figur