Meson Spectroscopy from Lattice QCD

Some recent progress in using lattice QCD to perform first-principles calculations of the spectra of mesons is discussed. In particular, I highlight some new results on resonances, near-threshold states and related scattering phenomena—this is a theoretically and experimentally interesting area where we have made significant advances in the last few years.

[L. Giusti, 2006] Cost of vacuum sampling The "noughties" have seen a lot of progress: Giusti, 2006] Dramatic change in scaling with quark mass from algorithm improvements [Hasenbusch '01,Lüscher '03,'04] Coupled to continued fall in price of CPU cores, means simulations at the physical quark masses are possible [PACS-CS arXiv:0911.2561] Excited-state energies measured from matrix of correlators: Solve generalised eigenvalue problem: Computing the spectrum (2) Excited-state energies measured from matrix of correlators: Solve generalised eigenvalue problem: Method constructs optimal ground-state creation operator, then orthogonal states

Isovector meson correlation functions
To create a meson, we need to build functions that couple to quarks. Meson can be created by a quark bilinear. Appropriate gauge invariant creation operator (for isospin I = 1) would be where Γ is some appropriate Dirac structure, and U C a product of (smeared) link variables. Operators that transform irreducibly under the lattice rotation group O h are needed. Complication: we do not have direct access to the fermion integration variables in the computer. The quark action is bilinear so: Now the elementary component in the correlation function is In general, this is still expensive to compute, since it requires knowing many entries in the inverse of the fermion operator, M.
If the choice of operator at the source is restricted and no momentum projection is made, only the bilinear at (eg) the origin on time-slice 0 is needed. Quark propagation from a single site to any other site is computed by solving Mψ = e a,α 0 where e 0 are the 12 vectors that only have non-zero components at the origin. Can we get away from this restriction?
Isovector meson correlation functions (3) The most general operator.
A restricted correlation function accessible to one point-to-all computation.

Isoscalar meson correlation functions
If we are interested in measuring isoscalar meson masses, extra diagrams must be evaluated, since four-quark diagrams become relevant. The Wick contraction yields extra terms, since Smearing -an essential ingredient for precision To build an operator that projects effectively onto a low-lying hadronic state need to use smearing Instead of the creation operator being a direct function applied to the fields in the lagrangian first smooth out the UV modes which contribute little to the IR dynamics directly. A popular gauge-covariant smearing algorithm; Jacobi/Wuppertal smearing: Apply the linear operator J = exp(σ∆ 2 ) ∆ 2 is a lattice representation of the 3-dimensional gauge-covariant laplace operator on the source time-slice

Redefine smearing
After tuning the free parameter σ it turns out J is a very low rank operator. The choice of smearing operator is arbitrary, provided 1 It is a scalar operator 2 It is gauge covariant 3 It is a function of only field on time-slice t (or perhaps a few nearest neighbours?) Redefine smearing to be a projection operator onto a low-dimensional space of fields: This is distillation.
How to choose v? One simple choice is to use the lowest M eigenvectors of ∆ 2 Distilled correlation functions Why is this helpful? Look at correlation functions such as an isovector meson two-point function Γ 1,2 are creation operators that make mesons with appropriate quantum numbers Inserting the definition of the distillation operator, the correlation function becomes a trace over a product of rank-M matrices.
The lowest eigenvector of the laplace operator Example: J PC = 2 ++ meson creation operator Need more information to discriminate spins. Consider continuum operator that creates a 2 ++ meson: Lattice: Substitute gauge-covariant lattice finite-difference D latt for D Example: J PC = 2 ++ meson creation operator Need more information to discriminate spins. Consider continuum operator that creates a 2 ++ meson: Lattice: Substitute gauge-covariant lattice finite-difference D latt for D A reducible representation: Example: J PC = 2 ++ meson creation operator Need more information to discriminate spins. Consider continuum operator that creates a 2 ++ meson: Lattice: Substitute gauge-covariant lattice finite-difference D latt for D A reducible representation: Look for signature of continuum symmetry:  Comparison is with data from Graz group Two's company -multi-meson states  Precision is crucial to go further.
New method show promise for more precision needed in the isoscalar meson sector.