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On lattice actions for static quarks

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Michele Della Morte, Andrea Shindler, Rainer Sommer

High Energy Physics - Lattice · Submitted 7 Jun 2005 · 53 references

Pith's verdict: Smeared links in the static propagator buy real time separation, the mechanism is checked against independent correlators, and the numbers other people need are all here.

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arXiv IDhep-lat/0506008 (version 1)
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CitationAndrea Shindler, Michele Della Morte, Rainer Sommer. "On lattice actions for static quarks." arXiv:hep-lat/0506008 (2005). https://pith.science/paper/CXB2CRO7
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The paper

The paper claims

Replacing the gauge link that carries a static quark with a smeared link suppresses the exponential growth of statistical noise in static-light lattice correlation functions, letting the signal reach roughly 2 fm.

The reading

Lattice calculations of B-meson quantities need static (infinitely heavy) quarks, but the standard Eichten–Hill discretization has a noise problem: the statistical error on a static-light correlation function grows exponentially with Euclidean time, and the growth rate diverges as the lattice spacing shrinks, so the signal is effectively lost beyond about 1 fm. The authors propose replacing the single temporal gauge link that transports the static quark by a smeared link — an APE-style staple average, an approximation to the SU(3) one-link integral, or a HYP link — and argue that this lowers the static quark's self-energy without changing the theory's continuum limit or its automatic $O(a)$ improvement. They check this in quenched simulations at $g_0^2=6$ and obtain a usable signal out to roughly 2 fm, with the HYP2 action best. They also supply the improvement coefficients and non-perturbative renormalization constants needed to actually use the new actions for the static-light axial current and for the b-quark mass in the static approximation.

What carries the argument

The construction is a generalized static action $S_h^W = a^4,%frac{1}{1+a%delta m_W}%sum_x %bar{%psi}_h (D_W^0+%delta m_W)%psi_h$, where the covariant derivative uses a gauge parallel transporter $W(x,0)$ in place of the temporal link $U(x,0)$; Eichten–Hill is the special case $W=U$. The static propagator is then an ordered product of these transporters, so the whole change is a change of the static quark's effective self-energy. The load-bearing identity is the noise formula $R_{NS}%propto %exp[(E^{stat}-(m_%pi+V(0))/2)x_0]$, which says the relevant quantity is not $E^{stat}$ alone but $E^{stat}-V(0)/2$ — this is why the APE action, whose $aE^{stat}=0.85(2)$ is larger than Eichten–Hill's $0.68(9)$, still wins, because its $aV(0)=0.671(2)$ is nonzero while Eichten–Hill's vanishes. The argument that no new improvement terms are needed rests on preserving heavy-quark spin symmetry and local heavy-flavour conservation, which keeps the actions in the same universality class. The improvement coefficients $c_A^{stat}$ and $b_A^{stat}$ are fixed by matching ratios of correlators between a reference action with known coefficients and the new action, then extrapolating in $a/L$.

Extended reading notes

Core claim

The paper's claim is that the exponential growth of the noise-to-signal ratio in static-light correlators is not universal: it is governed by $E^{stat}-V(0)/2$, the static quark's self-energy minus half the lattice heavy-quark potential at zero distance, and that quantity can be reduced by changing the parallel transporter $W$ inside the static action. The reasoning is that the variance of the correlator is dominated by a state containing a static quark–antiquark pair plus a light pair, with minimum energy approximated by $V(0)+m_%pi$, giving $R_{NS}%propto %exp[(E^{stat}-(m_%pi+V(0))/2),x_0]$; the linearly divergent piece of $E^{stat}$ is what drives the exponential, and smearing shrinks it. The one-loop coefficients $r^{(1)}$ support the ordering: $0.16845(2)$ for Eichten–Hill, $0.05737(4)$ for the APE and one-link-integral actions, $0.04844(1)$ and $0.03523(1)$ for the two HYP variants. A quenched run on $16^3%times 32$ at $%beta=6$ with nearly 5000 configurations confirms the predicted ordering and shows the measured $R_{NS}$ following the formula. Scaling tests in the Schrödinger functional show lattice artefacts that are small and linear in $a^2$ for all actions, and for the step scaling function $%Sigma_m$ of the b-quark mass they are visibly smaller with the HYP actions than with Eichten–Hill.

Load-bearing premise

The quantitative explanation of the improvement, and the expectation that the ordering of actions survives at finer lattice spacings, rests on the estimate that the dominant noise state has energy $V(0)+m_%pi$ — introduced as a plausible argument rather than derived, and not tested at lattice spacings finer than the ones simulated here.

Editorial extensions

If this is right

  • Static-light correlation functions become usable at Euclidean times of 1–2 fm, where the ground state can be isolated, rather than being swamped by noise beyond about 1 fm.
  • The improvement coefficients are now available for all four new actions, with $c_A^{stat}%approx 0.039(4)g_0^2$ and $0.220(14)g_0^2$ for HYP1 and HYP2, and $b_A^{stat}%approx 1/2+0.078(8)g_0^2$ and $1/2+0.259(13)g_0^2$ respectively.
  • The non-perturbative renormalization constants $Z_A^{stat}$ at $L_{ref}=1.436r_0$ are tabulated and parameterized in $%beta$ for each action, so bare matrix elements of the static-light axial current can be converted to the SF scheme at that scale.
  • The regularization-dependent pieces entering the static-approximation b-quark mass, $a%Gamma^{stat}(g_0,L_{ref}/a)$, are computed and parameterized, allowing the finite-volume step-scaling strategy to be repeated with the lower-noise actions.
  • Because lattice artefacts differ between the HYP1 and HYP2 actions, the authors recommend computing with both and constraining continuum extrapolations by universality.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The same logic should transfer to unquenched simulations: the noise formula contains no fermion-determinant physics, so the ordering of actions by $E^{stat}-V(0)/2$ ought to survive at $N_f=2$, which is where the payoff for B-physics would actually be collected.
  • If the exponential rate is set by a self-energy that can be engineered downward, one could imagine optimizing the smearing parameters directly against $r^{(1)}$ rather than against the signal, turning action choice into a small quantitative design problem.
  • The paper's own observation that $c_A^{stat}$ is largest for the action with the smallest $r^{(1)}$ suggests the two figures of merit — noise suppression and size of improvement coefficients — trade off, so a user choosing an action faces a genuine two-criterion decision rather than a single best choice.
Original abstract

We introduce new discretizations of the action for static quarks. They achieve an exponential improvement (compared to the Eichten-Hill regularization) on the signal to noise ratio in static-light correlation functions. This is explicitly checked in a quenched simulation and it is understood quantitatively in terms of the self energy of a static quark and the lattice heavy quark potential at zero distance. We perform a set of scaling tests in the Schroedinger functional and find scaling violations in the O(a) improved theory to be rather small -- for one observable significantly smaller than with the Eichten-Hill regularization. In addition we compute the improvement coefficients of the static light axial current up to O(g_0^4) corrections and the corresponding renormalization constants non-perturbatively. The regularization dependent part of the renormalization of the b-quark mass in static approximation is also determined.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 4,963 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.06 · states limitation 0.28 · human participants 0.01 · controlled comparison 0.84 · randomized assignment 0.01 · states quantitative result 0.26 · compares with prior results 0.98 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; .

We introduce new discretizations of the action for static quarks. They achieve an exponential improvement (compared to the Eichten-Hill regularization) on the signal to noise ratio in static-light correlation functions. This is explicitly checked in a quenched simulation and it is understood quantitatively in terms of the self energy of a static quark and the lattice heavy quark potential at zero distance. We perform a set of scaling tests in the Schroedinger functional and find scaling violations in the O(a) improved theory to be rather small -- for one observable significantly smaller than with the Eichten-Hill regularization. In addition we compute the improvement coefficients of the static light axial current up to O(g_0^4) corrections and the corresponding renormalization constants non-perturbatively. The regularization dependent part of the renormalization of the b-quark mass in static approximation is also determined.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 We introduce new discretizations of the action for static quarks. contribution 1.00 0.96 0.12 asserted 0.99
s1 They achieve an exponential improvement (compared to the Eichten-Hill regularization) on the signal to noise ratio in static-light correlation functions. result 0.98 0.86 0.56 asserted 1.00
s2 This is explicitly checked in a quenched simulation and it is understood quantitatively in terms of the self energy of a static quark and the lattice heavy quark potential at zero distance. result 0.53 0.93 0.22 asserted 1.00
s3 We perform a set of scaling tests in the Schroedinger functional and find scaling violations in the O(a) improved theory to be rather small -- for one observable significantly smaller than with the Eichten-Hill regularization. result 0.92 0.97 0.11 asserted 1.00
s4 In addition we compute the improvement coefficients of the static light axial current up to O(g_0^4) corrections and the corresponding renormalization constants non-perturbatively. contribution 0.61 0.97 0.06 asserted 1.00
s5 The regularization dependent part of the renormalization of the b-quark mass in static approximation is also determined. result 0.81 0.93 0.06 asserted 1.00

What kind of paper the classifier reports this is: method or tool 0.89 · evidence from simulation or computation 1.00.

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures
Figure 1. Schematic representation of the correlation functions f stat A (x0) and f stat 1 in the SF. The double lines indicate the static quark propagators. expects the large x0 asymptotic behavior f stat A (x0) ∝ e −Estatx0 , (2.9) where Estat is the binding energy of the static–light system. It depends on the choice for W and diverges approximately linearly in the lattice spacing. Estat ∼ Eself + O(a 0 ) ∼ 1 a e (1) g 2 0 + . . . . (2.10) This divergence is to be canceled by δmW = −Eself + O(a 0 ). The finite part of δmW is of course scheme dependent, but in perturbation th
Figure 2. The ratio RNS for the correlation function f stat A (x0) for a statistics of 5000 measurements on a 163 × 32 lattice at β = 6. Filled circles refer to S EH h , empty circles to S A h (S s h gives similar results), empty (filled) triangles to S HYP1 h (S HYP2 h ). The largest improvement is again given by the action S HYP2 h , but we see from the figure that all the proposed discretizations produce a clear signal (for the considered statistics) at least up to a time separation of roughly 2 fm. The dotted lines in figure 2 7
Figure 3. Numerical results for the improvement coefficient c stat A . For S EH h the result of [34] is plotted, for S s h the band represents the result eq. (3.4), while otherwise it refers to the numerical estimates eqs. (3.5, 3.6). Here the statistical errors are not the relevant ones. Rather our dominant error is due to the assumption (tested to some extent as just explained) that the 1-loop value for c stat A for action S 1 h = S A h is accurate for our values of g 2 0 . Apart from this, our determinations do provide a non-perturbative computation of c stat A for the other actions. Thus…
Figure 4. Numerical results for the improvement coefficient b stat A , presented as in figure 3. One comment is in order here. In the condition in eq. (3.7), O(amq)-terms with coefficients bg and bζh [21] are neglected, while terms proportional to bζ (see [40]) drop 11
Figure 5. Scaling plots for ξA(0, 0.5), ξ1(0, 0.5) and h(1/4). Symbols are as in figure 2, some of them have been shifted on the horizontal axis for clarity. Again the results for S s h are similar to the ones from S A h and haven’t been plotted. Here we simply define it as Σm(u, a/L) = 2L[Γstat(2L) − Γstat(L)]u=¯g 2(L) (3.14) Γstat(L) = 1 2a ln f stat A,I (x0 − a)/fstat A,I (x0 + a)
Figure 6. The step scaling function Σm for the different static actions. Symbols are chosen as in figure 2. Results for the Eichten–Hill action are taken from [9]. 4 Renormalization of the axial current and quark mass In the effective theory the static–light axial current is not derived from a symmetry transformation of the action. The renormalized current is therefore scale dependent. This dependence has been studied non-perturbatively, over a wide energy range, in the SF scheme (see [35]). The main quantity considered in that study is the step scaling function Σstat A (u, a/L) defined as Σ…
Figure 7. The step scaling function Σ(3.48, a/L). Symbols are as in figure 2, some of them have been shifted on the horizontal axis for clarity. The results for S s h are similar to those from S A h and haven’t been included in the plot. Data for the Eichten-Hill action and the continuum limit value have been taken from [35]. again we see that the different regularizations yield very similar results already for finite lattice spacing, although in principle agreement needs to be found only after having taken the continuum limit. At the scale Lref = 1.436r0 we have also computed the renormaliz
Figure 8. Dependence of ∆c, eq. (A.27), on a/L. of (θ,θ ′ ). It is clear that the limit a/L → 0 can be taken with reasonable precision. The plot indicates also that defining the improvement coefficient without performing the continuum limit will give a perturbative O(a/L) uncertainty of the order of 0.02 g 2 0 . To obtain the final number eq. (3.4) we used both the extrapolation procedures of [48] and [49], finding consistent results. A.5 Determination of b stat A To determine b stat A we follow the strategy just applied to compute c stat A . The ratio ρ˜I(Sh; m′ q , mq) = (1 + a
Figure 9. Dependence of ∆b, eq. (A.34), on a/L. groups SU(N) with N > 2, [51, 52]. A more convenient approximation can be inferred form the SU(2) case, where the group integral is proportional to V (x, 0) and takes the form U(x, 0) = V v I2(24v/g2 0 ) I1(24v/g2 0 ) , v =  1 N tr V V † 1/2 , V ≡ V (x, 0) , (B.2) in terms of the Bessel functions I1 and I2. For SU(3) we make a similar ansatz W(x, 0) = g −2 0 V (x, 0)s(g −2 0 v) . (B.3) Although this can not reproduce the SU(3) one-link integral exactly, since that is a more complicated function of V and V † , it turns out th

Figures reproduced from arXiv:hep-lat/0506008 by the authors.

References
How the literature uses it

52 arXiv papers cite this in 68 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 23 sentences without a confident label

Uses its method or tool 37 papers
  • That is, we are using a domain-wall action for the up, down, and strange quarks [ref] , the Iwasaki action [ref] for the gluons, and the Eichten-Hill action [ref] with HYP-smeared gauge links [ref] for the static heavy quark.

    $\Lambda_b \to p l^- \bar{\nu}$ form factors from lattice QCD with static b quarks · Lattice calculation · 1.00

  • We have combined a light doublet of twisted-mass fermions ($ψ^T = (u,d)$) defined at maximal twist with a static quark described by the HYP2 action [ref] to improve the signal-to-noise ratio [ref] : [equation] In order to extract the decay constant using maximally twisted lattice QCD, we need to evaluate the matrix element of the static-light local current.

    f_B and f_Bs with maximally twisted Wilson fermions · Heavy-light decay constant in the static limit of HQET · 1.00

  • The HQET observables themselves are computed using two different static actions, referred to as $$ and $$ [ref] , in order to have an improved noise-to-signal behaviour compared to the original Eichten-Hill action.

    Non-perturbative tests of continuum HQET through small-volume two-flavour QCD · Definitions · 0.97

Uses its data or values 3 papers
Cites it as support 1 paper
  • [ref] ).] Knowing the renormalization pattern of the non-local operator ([ref]), we wish to define [equation] as a completely power divergence free matrix element, [On the lattice, the exponential factor for the liner divergence subtraction should be $e^-ln(1+δm)|δz|$.] and hence we propose “power divergence subtracted” quasi quark distribution functions by [equation] which would make the theory well-defined.

    Practical quasi parton distribution functions · Renormalization of Wilson line and subtraction of linear UV divergence · 0.96

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