Simple check of the vacuum structure in full QCD lattice simulations

Given the increasing availability of lattice data for (unquenched) QCD with Nf = 2, it is worth while to check whether the generated vacuum significantly deviates from the quenched one. I discuss a specific attempt to do this on the basis of topological susceptibility data gained at various sea-quark masses, since for this observable detailed predictions are available. The upshot is that either discretization effects in dynamical simulations are still untolerably large or the vacuum structure in 2-flavour QCD substantially deviates from that in the theory with 3 (or 2+1) light quarks. INTRODUCTION In a historical perspective, the path towards phenomenological predictions of QCD by means of lattice techniques involves three steps: Pure Yang-Mills theory (where there is just glueball physics), quenched QCD (where the vacuum is the one in the SU(3) theory, but observables may involve so-called current -quarks) and full QCD (where the fermion determinant with the dynamical sea-quarks accounts for the quark loops in the vacuum). Today, the lattice community makes the final push towards full QCD, despite the fact that state-of-the-art simulations are modestly announced as “partially quenched” which means that the seaand current-quark masses in the (euclidean) generating functional Z[η̄,η] = ∫ DA e−SG ∏ Nf det(D/+msea) exp( ∫ η̄ 1 D/+mcur η) (1) are (in general) unequal and in most cases significantly heavier than the physical uand d-quarks, so that phenomenological statements require a twofold extrapolation. Since the finite sea-quark mass constitutes the key ingredient in this ultimate step (note that the determinant turns into a constant for msea → ∞, hence (1) reduces to the quenched generating functional in that limit), an obvious task is to check whether these “partially quenched” or “full” QCD simulations exhibit the change in the vacuum structure expected to occur if the fermions are “active” (i.e. if the back-reaction of the “dynamical” fermions on the gauge background is taken into account). The prime observable used to distinguish the respective vacua is the topological susceptibility χ(msea) = 〈q2〉 V , (2) with q the (global) topological charge, because detailed theoretical predictions show that χ behaves rather different in the quenched (msea →∞) and chiral (msea→0) limits, respectively. Even though in the lattice-regulated theory (and with certain definitions of the topological charge operator) q may be somewhat ambiguous on the level of a single configuration, the moment of the q-distribution which enters (2) can be measured with controlled error-bars, and as a purely gluonic object the resulting χ =χ(msea) encodes nothing but the vacuum structure of the theory. Below, I give a quick survey of recent lattice determinations of χ at various sea-quark masses in Nf =2 QCD, I discuss the available continuum knowledge of the functional form χ = χ(msea), and I present a non-standard lattice determination of the quenched topological susceptibility χ∞ and the chiral condensate Σ based on it. The outcome is that either certain observables in todays phenomenological studies with light dynamical quarks suffer from large discretization effects or – the more speculative view – that the low-energy structure of QCD with Nf =2 is substantially different from that with Nf =3. LATTICE DATA I start with a quick survey of recent lattice data for the topological susceptibility in QCD with 2 dynamical flavours; the selection reflects nothing but my personal awareness. CP-PACS: The CP-PACS collaboration has simulated full QCD on several grids at various (β,κ)-values, using an RG-improved gauge action and an O(a)-improved fermion action with mean-field values for the associate cSW coefficients. Below, I concentrate on the data generated on the 243 ×48 lattice at β=2.1 with LW-cooling [1]. UKQCD: The UKQCD collaboration has simulated full QCD on a 163 × 32 grid at various (β,κ)-values, using the standard (Wilson) gauge action and an O(a)-improved fermion action with the non-perturbative values for the associate cSW coefficients [2]. SESAM/TXL: The SESAM/TXL collaboration has simulated full QCD on two grids (163 × 32 and 243 × 40) at several (β,κ)-values, combining the unimproved (Wilson) gauge action with unimproved (Wilson) fermions (i.e. setting cSW =0) [3]. Thin link staggered: Trusting a continuum identity for the relationship between the 2and the 4-flavour functional determinant, the staggered fermion action may be used to simulate QCD with Nf =2. There are data by the Pisa group [4] and by A. Hasenfratz based on configurations by the MILC collaboration and the Columbia/BNL project [5]. Fig. 1 displays the data, along with continuum constraints to be discussed next. CONTINUUM KNOWLEDGE As mentioned in the introduction, the data for the topological susceptibility χ versus the sea-quark mass m≡msea prove useful to test the vacuum structure, because continuum QCD provides us with detailed predictions: There are analytic upper bounds for χ(m) at both asymptotically small and large sea-quark masses and there is a “semi-analytic” formula for χ(m) valid at intermediary quark masses (where the bulk of the lattice data reside). The only caveat is that these bounds hold true in the continuum limit, but so far no continuum extrapolation for χ(m) in Nf =2 QCD is available yet. Before stating the complications due to this, the continuum functional forms shall be discussed. 0 0. 01 0. 02 0. 03 0. 04 0. 05 0. 06 0. 07 0. 08 0 1 2 3 4 5 6 7 8 χ r 0 4 (M π r 0 ) 2 T op . S us c. W ils on -t yp e (C P -P A C S ,U K Q C D ,S E S A M ) vs . q ua rk m as s