Friday, January 29, 2010

Excited states from the lattice, 1 of n

This post is intended as the first in a series about techniques for the extraction of information on excited states of hadrons from lattice QCD calculations.

As a reminder, what we measure in lattice QCD are correlation functions C(t)=<O(t)O(0)> of composite fields O(t). From Feynman's functional integral formula, these are equal to the vacuum expectation value of the corresponding products of operators. Changing from the Heisenberg to the Schrödinger picture, it is straightforward to show that (for infinite temporal extent of the lattice) these have a spectral representation C(t)=Σnn|2 e-Ent, which in principle contains all information about the energies En and matrix elements ψn=<0|O|n> of all states in the theory.

The problem with getting that information from the theory is twofold: Firstly, we only measure the correlator on a finite number of timeslices; the task of inferring an infinite number of En and ψn from a finite number of C(tk) is therefore infinitely ill-conditioned. Secondly, and more importantly, the measured correlation functions have associated statistical errors, and the number of timeslices on which the excited states' (n>1) contributions are larger than the error is often rather small. We are therefore faced with a difficult data analysis task.

The simplest idea of how to extract information beyond the ground state would be to just perform a multi-exponential fit with a given number of exponentials on the measured correlator. This approach fails spectacularly, because multi-exponential fits are rather ill-conditioned. One finds that changing the number of fitted exponentials will affect the best fit values found rather strongly, leading to a large and unknown systematic error; moreover, the fits will often tend to wander off into unphysical regions (negative energies, unreasonablely large matrix elements for excited states). This instability therefore needs addressing if one wishes to use a χ2-based method for the analysis of excited state masses.

The first such stabilisation that has been proposed and is widely used is known as Bayesian or constrained fitting. The idea here is to augment the χ2 functional by prior information that one has about the spectrum of the theory (such as that energies are positive and less than the cutoff, but if one wishes also perhaps more stringent constraints coming e.g. from effective field theories or models). The reason one may do this is Bayes' theorem, which can be read as stating that the probability distribution of the parameters M given the data D is the product of the probability distribution of the data given the parameters times the probability distribution of the parameters absent any data: P(M|D)=P(D|M)/P(D) P(M); taking the logarithm of both sides and maximising of M, we then want to maximise log(P(D|M)) + log(P(M)). Now log(P(D|M)) is known to be proportional to 2, so if P(M) was completely flat, we would end up minimizing χ2. If we take P(M) to be Gaussian instead, we end up with an augmented χ2 that contains an additional term Σn (Mn-In)2n2 that forces the parameters Mn towards their initial guesses ("priors") In, and hence stabilises the fit -- in principle even with an infinite number of fit parameters. The widths σn are arbitrary in principle; fitted values Mn that noticeably depend on σn are determined by the priors and not the data and must be discarded. In practice the lowest few energies and matrix elements do not show a significant dependence on σn or on the number of higher states included in the fit, and may therefore be taken to have been determined by the data.

Bayesian fitting is a very powerful tool, but not everyone is happy with it. One objection is that adding any external information, even as a constraint, compromises the status of lattice QCD as a first-principles determination of physical quantities. Another common worry is the GIGO (garbage in-garbage out) principle with regards to the priors.

A way to address the former concern that has been proposed is the Sequential Empirical Bayes Method (SEBM). Here, one first performs an unstabilised single-exponential fit at large times t, where the ground state is known to dominate. Then one performs a constrained two-exponential fit over a larger range of t using the first fit result as a prior (with its error as the width). The result of this fit is then used as the prior in another three-exponential fit over an even larger time range, and so forth. (There is some variation as to the exact procedure followed, but this is the basic idea). In this way, all priors have been determined by the data themselves.

In the next post of this series we will look at a completely different approach to extracting excited state masses and matrix elements that does not rely on χ2 at all.

Friday, January 08, 2010

New book on the lattice

There was a time when the only textbooks on lattice QCD were Montvay&Münster and Creutz. Not so any more. Now the new textbook "Quantum Chromodynamicson the Lattice: An Introductory Presentation" by Christof Gattringer and Christian Lang (Lecture Notes in Physics 788, Springer) offers a thorough and accessible introduction for beginners.

Gattringer and Lang start from a derivation of the path integral in the context of Quantum Mechanics, and after deriving the naive discretisation of lattice fermions and the Wilson gauge action present first the lattice formulation of pure gauge theory, including the Haar measure and gauge fixing, with Wilson and Polyakov loops and the static quark potential as the observables of interest. Numerical simulation techniques for pure gauge theory are discussed along with the most important data analysis methods. Then fermions are introduced properly, starting from the properties of Grassmann variables and a discussion of the doubling problem and the Wilson fermion action, followed by chapters on hadron spectroscopy (including some discussion of methods for extracting excited states), chiral symmetry on the lattice (leading through the Nielsen-Ninomiya theorem and the Ginsparg-Wilson relation to the overlap operator) and methods for dynamical fermions. Chapters on Symanzik improvement and the renormalisation group, on lattice fermion formulations other than Wilson and overlap, on matrix elements and renormalisation, and on finite temperature and density round off the volume.

The book is intended as an introduction, and as such it is expected that more advanced topics are treated briefly or only hinted at. Whether the total omission of lattice perturbation theory (apart from a reference to the review by Capitani) is justified probably depends on your personal point of view -- the book clearly intends to treat lattice QCD as a fully non-perturbative theory in all respects. There are some other choices leading to the omission or near-omission of various topics of interest: The Wilson action is used both for gluons and quarks, although staggered, domain wall and twisted mass fermions, as well as NRQCD/HQET, are discussed in a separate chapter. The calculation of the spectrum takes the front seat, whereas the extraction of Standard Model parameters and other issues related to renormalisation are relegated to a more marginal position.

All of these choices are, however, very suitable for a book aimed at beginning lattice theorists who will benefit from the very detailed derivations of many important relations that are given with many intermediate steps shown explicitly. Very little prior knowledge of field theory is assumed, although some knowledge of continuum QFT is very helpful, and a good understanding of general particle physics is essential. The bibliographies at the end of each chapter are up to date on recent developments and should give readers an easy way into more advanced topics and into the research literature.

In short, this book is a gentle, but thorough introduction to the field for beginners which may also serve as a useful reference for more advanced students. It definitely represents a nice addition to your QCD bookshelf.