1. Limitations of Partial Quenching Stephen Sharpe and Ruth Van de Water, University of Washington, Seattle QCD with 3 flavors (u, d, s) possesses an approximate SU(3) chiral symmetry This can be used to construct an effective Lagrangian in terms of the low-energy degrees of freedom – the pseudoscalar mesons (π’s, K’s, η): Chiral Perturbation Theory (χPT) Partially Quenched (PQ) Lattice QCD Partially Quenched χPT Determining L PQ New PQ Operator’s Contributions to Meson Masses and Decay Constants Analytic NNLO Contributions to Meson Masses from L 6 Operators Analytic NNLO Contributions to Meson Masses from L 6 Operators ConclusionsConclusions χPT is a perturbative expansion in both and The O (p 2, m π 2 ) Lagrangian is: and the O (p 4, m π 4 ) (Gasser-Leutwyler) Lagrangian is: (Here a linearly-dependent operator was eliminated using SU(3) group relations) Lattice QCD allows calculation of these coefficients from first principles, if one can simulate with sufficiently light quarks A practical obstacle is the computational expense of lower sea- quark masses A cheaper alternative is to lower valence-quark masses at fixed moderately light sea-quark masses (the Partially Quenched Approximation) Although PQ QCD is unphysical, the properties of pseudoGoldstone mesons can be expressed in terms of the physical coefficients of the QCD chiral Lagrangian through next-to-leading order (NLO) in χPT 1,2 Here we show that this does not hold at next-to-next-to- leading order (NNLO) Partial quenching introduces unphysical bosonic ghost quarks 3 Therefore partial quenching changes the low-energy symmetry structure– the chiral symmetry group becomes a graded Lie Algebra with both commutation and anti- commutation relations: This operator contributes to unphysical scattering processes involving ghost quarks at NLO and affects meson masses and decay constants at NNLO To simplify calculations we considered only two valence quarks, A and B, and N sea quarks of equal mass The charged meson, π AB, receives NNLO corrections to its renormalized mass from the following loop diagram: It also receives NNLO corrections to its decay constant from the following diagrams, the first of which renormalizes the meson wavefunction, and the second of which modifies the axial current: All of these diagrams receive contributions from the following classes of quark diagrams, which illustrate the flow of quarks within the mesons: We determined the following corrections to the renormalized meson mass and decay constant due to the new operator (using dimensional regularization and MS): (Quark line diagrams show the effect of partial quenching – valence quarks appear only as external states because ghost quarks cancel them in loops ) Because of the graded group structure a new operator exists in L 4 : Like f π and L 1 to L 8, L PQ can be obtained by fitting to lattice experiments To illustrate this we calculated meson scattering processes that only receive contributions from double Supertrace operators in PQχPT, such as: L 1, L 2, and L PQ can, in principle, be separately determined by using amplitudes for three such processes in PQχPT and fitting them to lattice data This amplitude vanishes at LO The O (p 4 ) terms in the scattering amplitude are: = Fits of present PQ lattice data require NNLO terms 4,5 Full non-analytic NNLO calculation for PQ QCD is not available; however, as an intermediate step we have determined the form of the analytic NNLO terms Mesons get NNLO corrections to their masses from operators in L 6 from the diagram: O (p 6,m 6 ) Over a dozen O (p 6,m 6 ) operators contribute to this diagram, such as: Fits to PQ data with 1/3m S < m Sea < 2/3m S are consistent with this 4,5 We expect a similar expression to hold for f AB PQχPT can be used in conjunction with PQ Lattice QCD to determine the low-energy constants of real QCD However, partial quenching changes the chiral symmetry group, and introduces an new linearly-independent O (p 4 ) operator This operator adds corrections to the meson masses and decay constants at NNLO in PQχPT These corrections can be determined, in principle, by calculating appropriate scattering amplitudes In this way, the program of using unphysical PQ simulations to determine physical parameters could be extended to NNLO (Here “V” and “S” indicate valence and sea quarks in the loop meson) (Here mesons are written in the quark basis, not the usual generator basis) A convenient linear combination is one that vanishes in the unquenched SU(3) limit: [1] S. Sharpe and N. Shoresh, Phys. Rev. D 62, 094503 (2000). [2] S. Sharpe and N. Shoresh, Phys. Rev. D 64, 114510 (2001) [3] C.W. Bernard and M.F.L. Golterman, Phys. Rev. D 49, 486 (1994). [4] F. Farchoni et al, hep-lat/0307002 [5] F. Farchoni et al, hep-lat/0302011 ReferencesReferences χ is defined such that at lowest order χPT allows analytic calculations of low-energy QCD processes, but the results are in term of undetermined coefficients, f π and L 1 to L 8, which must be determined experimentally However, we found that only four linear combinations of quark masses appear in the NNLO tree-level correction to an AB meson’s mass: (Here α i are combinations of NNLO constants)