Frequencysplitting estimators of singlepropagator traces
Abstract
Singlepropagator traces are the most elementary fermion Wick contractions which occur in numerical lattice QCD, and are usually computed by introducing randomnoise estimators to profit from volume averaging. The additional contribution to the variance induced by the random noise is typically orders of magnitude larger than the one due to the gauge field. We propose a new family of stochastic estimators of singlepropagator traces built upon a frequency splitting combined with a hopping expansion of the quark propagator, and test their efficiency in twoflavour QCD with pions as light as 190 MeV. Depending on the fermion bilinear considered, the cost of computing these diagrams is reduced by one to two orders of magnitude or more with respect to standard randomnoise estimators. As two concrete examples of physics applications, we compute the disconnected contributions to correlation functions of two vector currents in the isosinglet \(\omega \) channel and to the hadronic vacuum polarization relevant for the muon anomalous magnetic moment. In both cases, estimators with variances dominated by the gauge noise are computed with a modest numerical effort. Theory suggests large gains for disconnected three and higher point correlation functions as well. The frequencysplitting estimators and their spliteven components are directly applicable to the newly proposed multilevel integration in the presence of fermions.
1 Introduction
Disconnected fermion Wick contractions contribute to many physics processes at the forefront of research in particle and nuclear physics: the hadronic contribution to the muon anomalous magnetic moment, \(K\rightarrow \pi \pi \) decays, nucleon form factors, quantum electrodynamics and strong isospinbreaking contributions to hadronic matrix elements, \(\eta '\) propagator to name a few. When computed numerically in lattice Quantum Chromodynamics (QCD) and if the distances between the disconnected pieces are large, their variances are dominated by the vacuum contribution. The latter are well approximated by the product of variances of the connected subdiagrams the contractions are made of. The recentlyproposed multilevel Monte Carlo integration in the presence of fermions [1, 2] is particularly appealing for computing disconnected contractions, since the various subdiagrams can be computed (essentially) independently from each other, thus making the scaling of the statistical error with the cost of the simulation much more favorable with respect to the standard Monte Carlo integration.
The simplest examples of this kind are the disconnected Wick contractions of fermion bilinear twopoint correlation functions, where each singlepropagator trace is usually computed by introducing randomnoise estimators [3, 4, 5]. As the action of the auxiliary fields is already factorized, the multilevel integration in the gauge field becomes highly profitable once the variance of each connected subdiagram is driven by its intrinsic gauge noise. The randomnoise contribution, however, is typically orders of magnitude larger than the one due to the gauge field, a fact which calls for more efficient estimators in order to avoid the need of averaging over many randomnoise fields with large computational cost.
The aim of this paper is to fill this gap by introducing a new family of stochastic estimators of singlepropagator traces which combine the newly introduced spliteven estimators with a frequency splitting and a hopping expansion of the quark propagator. We test their efficiency by simulating twoflavour QCD with pions as light as 190 MeV. As a result, depending on the fermion bilinear considered, the cost of computing singlepropagator traces is reduced by one to two orders of magnitude or more with respect to the computational needs for standard randomnoise estimators. The frequencysplitting estimators can be straightforwardly implemented in any standard Monte Carlo computation of disconnected Wick contractions, as well as directly combined with the newly proposed multilevel integration in the presence of fermions.
In the next section we summarize basic facts about variances of generic disconnected Wick contractions, while those of singlepropagator traces are discussed in Sect. 3. The following section is dedicated to introduce stochastic estimators of singlepropagator traces of heavy quarks based on a hopping expansion of the propagator, while in Sect. 5 we introduce the spliteven estimators for the difference of two singlepropagator traces also relevant for the muon anomalous magnetic moment. The frequencysplitting estimators are introduced in Sect. 6, where also the outcomes of their numerical tests are reported. In Sect. 7 we discuss the impact of these findings on two concrete examples of physics applications: the disconnected contributions to the correlator of two electromagnetic currents in the isospin limit relevant for the hadronic contribution to the muon anomalous magnetic moment, and the propagator of the \(\omega \) vector meson. The paper ends with a short section of conclusions and outlook, followed by some appendices where some useful notation and formulas are collected.
2 Variances of disconnected Wick contractions
Maybe the simplest example of this kind is a disconnected Wick contraction of the correlator of two bilinear operators for which, following Eq. (2.3), the variance is well approximated by the product of variances of two singlepropagator trace estimators.
3 Singlepropagator traces
3.1 Randomnoise estimator
Overview of the ensembles and statistics used in this study. We give the label, the spatial extent of the lattice, the hopping parameter \(\kappa \), the number of MDUs simulated after thermalization, the number of independent configurations selected \(N_{\mathrm{cfg}}\), the pion mass \(M_\pi \), and the product \(M_\pi L\). For F7, \(N_{\mathrm{cfg}}=100\) configurations have been used for estimating the variances while the final results for the twopoint functions have been obtained with \(N_{\mathrm{cfg}}=1200\)
id  L / a  \(\kappa \)  MDU  \(N_{\mathrm{cfg}}\)  \(M_\pi \) (MeV)  \(M_\pi L\) 

E5  32  0.13625  12, 800  100  440  4.7 
F7  48  0.13638  9, 600  \(100\,(1200)\)  268  4.3 
G8  64  0.136417  820  25  193  4.1 
3.2 Numerical tests
The first primary observables that we have computed are the estimators in Eq. (3.10) with Gaussian random noise. Their variances are shown in Fig. 1 as a function of the number of randomnoise sources \(N_s\) for the ensemble F7. Data for the E5 and the G8 lattices show the same qualitative behaviour. Variances go down linearly in \(1/N_s\) until the randomnoise contribution becomes negligible, see Eq. (3.11), after which a plateau corresponds to the gauge noise (dashed lines). The first clear message from the data is that the randomnoise contribution to the variances is comparable for the various bilinears, as suggested by Eq. (3.11), and it is orders of magnitude larger than the gauge noise. Moreover, the latter can vary by orders of magnitude among the various bilinears, see Sect. 6, with the densities having the largest gauge noise while the currents the smallest one.
4 Hopping expansion of singlepropagator traces
4.1 Numerical tests
We have computed the singlepropagator trace estimators \({{\bar{\tau }}}_{_{\Gamma ,r}}\), \({\bar{t}}^{M}_{_{\Gamma ,r}}\) and \({{\bar{\tau }}}^{R}_{_{\Gamma ,r}}\) for \(n=2\) on all ensembles listed in Table 1 for several valence quark masses. For F7 and for the subtracted bare quark mass \(a m_{q,r}=0.3\), the variances are shown in Fig. 2 for the pseudoscalar density and for a spatial component of the vector current respectively. Similar results are obtained for other bilinears and/or for the E5 and the G8 lattices. Furthermore, the variances are in the same ballpark as the free theory values computed using the results of Appendix B.3.
A clear picture emerges: the bulk of the randomnoise contribution to \(\sigma ^2_{{{\bar{\tau }}}{_{\Gamma ,r}}}\) is due to \(M_{2n,m_r}\) for all bilinears. Once the latter is subtracted from the propagator and its contribution to \({\bar{t}}{_{_{\Gamma ,r}}}\) is computed exactly, the random noise is reduced by approximately one order of magnitude or more. Notice that \(\sigma ^2_{{\bar{t}}^{M}_{_{\Gamma ,r}}}\) is from 2 (pseudoscalar) up to 5 (vector) orders of magnitude smaller than \(\sigma ^2_{{{\bar{\tau }}}{_{\Gamma ,r}}}\) for \(N_s=1\).
In Fig. 3, \(\sigma ^2_{{{\bar{\tau }}}^{R}_{_{\Gamma ,r}}}\) for \(N_s=1\) and \(\sigma ^2_{{\bar{t}}^{M}_{_{\Gamma ,r}}}\) multiplied by 10 both for \(n=2\) are shown as a function of the valence bare subtracted quark mass \(am_{q,r}\) for the pseudoscalar density and the spatial component of the vector current. As expected the variance reduction due to the subtraction of \(M_{2n,m_r}\) gets larger and larger at heavier quark masses. In particular at \(am_{q,r}=0.3\) the variance of the remainder is approximately one order of magnitude smaller than at the sea quark mass value of \(am_{q,r}=0.00207\). It is worth noting that even at this light mass, the randomnoise contribution to \(\sigma ^2_{{{\bar{\tau }}}{_{\Gamma ,r}}}\) from \(M_{2n,m_r}\) is still significant for all bilinears. The variance reduction due to HPE, however, is only a factor 2 or so.
All in all data suggest that at heavy masses an efficient estimator of \(s_{_{_{\Gamma ,r}}}\) is obtained by computing \({\bar{t}}^{M}_{_{\Gamma ,r}}\) exactly and the remainder via the stochastic estimator \({{\bar{\tau }}}^{R}_{_{\Gamma ,r}}\). Which is the optimal order n and how many random sources \(N_s\) are required for the remainder depend on the bilinear considered and on the final target observable of interest, see Sect. 6.
5 Differences of singlepropagator traces
5.1 Standard randomnoise estimator
5.2 Spliteven randomnoise estimator
The randomnoise contributions to the variances of the spliteven estimators in Eq. (5.9) are thus expected to be significantly smaller than for the standard estimators^{5} of differences of singlepropagator traces. This is not surprising since in this case both sources, \(\eta _i\) and \(\eta _i^\dagger \), are ultraviolet filtered by a quark propagator and the variance has one integral less in the timecoordinate analogously to the case of timediluted sources.^{6} With respect to the gauge variance, however, the randomnoise contribution is still expected to be larger.
5.3 Numerical tests
We have computed the two randomnoise estimators in Eqs. (5.6) and (5.9) on all ensembles listed in Table 1 and for several pairs of quark masses. For F7 and for the bare valence masses \(am_{q,r}=0.00207\) and \(am_{q,s}=0.0189\), corresponding to the sea and approximately the strange quark masses [9, 16], the variances are shown in Fig. 4 for the pseudoscalar density and for one spatial component of the vector current. The variance of the standard estimators \(\sigma ^2_{{{\bar{\theta }}}_{_{\Gamma ,rs}}}\) (red filled symbols) turns out to be essentially \(\Gamma \)independent as suggested by Eq. (5.7), and it is dominated by the spectral sum \(\langle SPSP\rangle \). The spliteven estimators \({{\bar{\tau }}}_{_{\Gamma ,rs}}(x_0)\) have much smaller variances.^{7} The reduction factor ranges from approximately one order of magnitude for the scalar and pseudoscalar densities up to around two orders of magnitude or more for the axial and vector currents as well as for the tensor bilinear. A similar reduction in the variance using the spliteven estimator is observed for ensembles with different seaquark masses, see Fig. 9 in Appendix D for the analogous figures for the E5 and G8 ensembles. The gauge noise is still smaller than the random noise, but with the spliteven estimator the number \(N_s\) of random sources needed to approach the gauge noise is moderate. It ranges from a few for the pseudoscalar density up to O(100) for the vector current.
The total cost of one evaluation of the FS estimator normalized to the cost of one evaluation of the standard estimator, for examples of the FS estimators on all ensembles. The cost of the spliteven components, \(\tau _{_{\Gamma ,r_kr_{k+1}}}\), is defined as the time for the two required inversions per source, while the cost of the remainder, \(\tau ^{_R}_{_{\Gamma ,r_m}}\), is the single inversion required per source. The standard estimator requires one inversion at the seaquark mass \(m_{q,r_0}\) per source. The relative cost of each component is given with the number of times it is evaluated for each full evaluation of the FS estimator, along with the bare subtracted quark masses which define the estimator. Note that the overhead required to compute the exact hopping terms is not included, which is performed only once per configuration. For \(n=2\) HPE this quickly becomes negligible when the FS estimator is evaluated multiple times
Ensemble  Estimator  k  \(am_{q,r_k}\)  Component  \(N_s\)  Rel. \(\hbox {cost/}N_s\) 

E5  FS1  0  0.00557  \(\tau _{_{\Gamma ,r_0r_1}}\)  1  1.42 
1  0.08  \(\tau _{_{\Gamma ,r_1r_2}}\)  3  0.66  
2  0.3  \(\tau ^{_R}_{_{\Gamma ,r_2}}\)  10  0.25  
Total rel. cost  5.81  
F7  FS1  0  0.00207  \(\tau _{_{\Gamma ,r_0r_1}}\)  1  1.26 
1  0.1  \(\tau ^{_R}_{_{\Gamma ,r_1}}\)  4  0.26  
Total rel. cost  2.30  
FS2  0  0.00207  \(\tau _{_{\Gamma ,r_0r_1}}\)  1  1.49  
1  0.02  \(\tau _{_{\Gamma ,r_1r_2}}\)  1  0.80  
2  0.06  \(\tau _{_{\Gamma ,r_2r_3}}\)  2  0.52  
3  0.15  \(\tau _{_{\Gamma ,r_3r_4}}\)  3  0.36  
4  0.3  \(\tau ^{_R}_{_{\Gamma ,r_4}}\)  10  0.15  
Total rel. cost  5.92  
G8  FS1  0  0.00108  \(\tau _{_{\Gamma ,r_0r_1}}\)  1  1.33 
1  0.01935  \(\tau _{_{\Gamma ,r_1r_2}}\)  1  0.51  
2  0.1  \(\tau ^{_R}_{_{\Gamma ,r_2}}\)  8  0.18  
Total rel. cost  3.28 
6 Frequencysplitting of singlepropagator traces
6.1 Numerical tests

FS1 is the simplest frequencysplitting estimator with one mass difference only. The masses are \(a m_q=0.00207\) and 0.1, \({{\bar{\tau }}}_{_{\Gamma ,r_0 r_{1}}}\) and \({{\bar{\tau }}}^{R}_{_{\Gamma ,r_1}}\) are defined with \(N_s=1\) and 4 respectively per full evaluation of the estimator. Each evaluation of this estimator costs approximately 2.5 times more than evaluating one random source for the standard estimator.^{9} See Table 2 for a detailed comparison of the relative cost of the estimators for our particular implementation.

FS2 is defined by 4 mass splittings corresponding to the masses \(a m_q=0.00207\), 0.02, 0.06, 0.15, 0.3, and the corresponding randomnoise estimators are defined with \(N_s=1\), 1, 2, 3, and 10 random sources respectively. Each application of this estimator costs approximately 6 times with respect to evaluating one random source for the standard estimator, see Table 2.
In Fig. 5 we show the variances of FS1, FS2 and of the standard estimator as a function of \(N_s\), the number of evaluations of each of them per gauge configuration. A clear message emerges: a large gain is obtained for both frequencysplitting estimators with mild differences in efficiency between them. The FS1 is slightly better for the scalar and pseudoscalar densities, while FS2 is more efficient for the vector, axialvector and tensor bilinears. In particular, the variance of FS1 is approximately 20 and 15 times smaller than the one of the standard estimators for the scalar and pseudoscalar densities respectively. Taking into account that one application of FS1 costs approximately 2.5 more, the gain in computation cost is 8 and 6 for the scalar and pseudoscalar^{10} densities respectively. For the vector and the axialvector, the variance of FS2 is approximately 2 orders of magnitude smaller than the one of the standard estimators. As the FS2 is 6 times more expensive, the gain in computational cost is approximately a factor 15. For the tensor the factor gained reaches approximately 20. Figure 10 of Appendix D depicts examples of the frequencysplitting estimators for the E5 and G8 ensembles, which illustrate similar orders of magnitude of improvement. In the vector channel, the improvement is even greater closer to the physical point.
7 Numerical tests for twopoint functions
In this section we discuss the numerical results for two representative examples of disconnected contributions to twopoint functions, which are the simplest correlation functions with a nontrivial time dependence composed only of singlepropagator traces. We use the estimators proposed in Sects. 5 and 6 to confirm the expected improvement over the standard estimator, and check the factorization formula for the variance given in Sect. 2.
7.1 Spliteven estimator for electromagnetic current
In the lefthand panel of Fig. 6, we show the variance of this correlation function for \(x_0/a=10\) computed by using the standard (red filled squares) and spliteven estimators (blue open squares) in Eqs. (5.6) and (5.9) respectively. A reduction of the variance of up to four orders of magnitude is obtained with the spliteven estimator (two orders of magnitude in the cost), which starts to be comparable to the gauge noise for \(N_s\sim 256\). As expected, the variance is practically constant in \(x_0\) and welldescribed by the factorization formula in Eq. (2.3) when the averaging over time and the polarizations of the current are taken into account.
In the righthand panel of Fig. 6 our best estimate of the correlation function using the spliteven estimator is shown using an increased number of gauge configurations, with respect to those used for estimating the variances, of \(N_{\mathrm {cfg}}=1200\). This in turn corresponds to a relative statistical precision of approximately \(10\%\) to the disconnected lightquark part of the muon anomalous magnetic moment coming from contributions to the integral up to timedistances of 1.5 fm. If the integral is computed up to 3.0 fm or so, the relative statistical error grows up to \(70\%\), calling for the multilevel integration to determine the contribution from the long distance part of the integrand. To properly renormalize the correlator each current has to be multiplied by the factor \(Z_V\) which brings a negligible error with respect to the statistical error of the bare correlator.^{11}
7.2 Frequencysplitting estimator for isoscalar vector currents
To evaluate this correlation function, we use the FS2 estimator introduced in Sect. 6 for both singlepropagator traces. In the left plot of Fig. 7 we show the variances of the standard estimator (filled symbols) against the number of sources, and the improved FS2 estimator (open symbols) against the number of its evaluations per gauge configuration. The gauge variance is approached with about \(N_s\sim 256\) evaluations of the FS2 estimator, similarly to the case of the onepoint function of Sect. 6. In this case, while the disconnected piece gives only a small contribution to the isoscalar channel at intermediate hadronic distances, its variance quickly dominates the statistical error at large distances. The improved estimator thus allows the full correlation function to be resolved at much larger distances.
8 Conclusions
The numerical computation of disconnected Wick contractions is challenging in lattice QCD because (a) their variances are dominated by the vacuum contribution, which in turn implies that statistical errors remain constant with the distance of the disconnected pieces while the signal typically decreases exponentially, and (b) averaging each disconnected subdiagram over the volume tends to be numerically expensive because the quark propagators must be recomputed at each lattice point.
A milestone for solving the second problem was the introduction of randomnoise estimators [3, 4, 5] which allow one to sum over many or all source points stochastically. However for singlepropagator traces, the simplest among the disconnected subdiagrams, such estimators tend to have variances which are typically orders of magnitude larger than the intrinsic gauge noise. An a priori theoretical analysis of the variances is thus mandatory for deciding how to define exactly the stochastic observables.
Luckily the randomnoise contribution to the variances can be reexpressed in the form of simple integrated correlation functions of local composite operators, a fact which allows us to use the quantum field theory machinery for analyzing the origin of the statistical errors and eventually to reduce them.
As a result, we have introduced new stochastic observables for singlepropagator traces: the spliteven and the frequencysplitting estimators for difference of two traces and for single traces respectively. The former needs from a few random sources for the pseudoscalar density up to O(100) for the vector current to approach the gauge noise. The reduction in numerical cost with respect to the standard estimator ranges from one order of magnitude for the scalar and pseudoscalar densities up to around two orders of magnitude or more for the axial and vector currents as well as for the tensor bilinear. Just one or a few evaluations of the frequencysplitting estimators are needed for the variances of the scalar, pseudoscalar and tensor bilinears to be comparable to the gauge noise, while for the axialvector and vector currents O(10) and O(100) evaluations are required to reach the same goal. In this case the reduction of the computational cost with respect to the standard estimator is of one order of magnitude or so depending on the bilinear. In all cases considered the variances of the stochastic estimators reach the level of the intrinsic gauge noise with a moderate number of evaluations per gauge configuration.
The use of these new estimators significantly speeds up the computation of disconnected fermion Wick contractions which contribute to many physics processes at the forefront of research in particle and nuclear physics: the hadronic contribution to the muon anomalous magnetic moment, \(K\rightarrow \pi \pi \) decays, nucleon form factors, quantum electrodynamics and strong isospinbreaking contributions to hadronic matrix elements, \(\eta '\) propagator, etc. As an example we have shown their potential for computing the disconnected contribution to the lightquark contribution to the muon anomalous magnetic moment and to the correlator of two singlet vector currents. Theory suggests large gains for disconnected three and higher point correlation functions as well. To solve or mitigate the problem (a) alluded to at the beginning of this section, the next step is to combine these estimators with the newly proposed multilevel integration in the presence of fermions [1, 2].
Footnotes
 1.
Without loss of generality we assume \(W_i(x)\) to be real.
 2.
Throughout this paper we focus on zero threemomentum fields only. All techniques presented, however, are directly applicable to fields with nonzero three momentum.
 3.
If not present in the theory, a valence quark \(\psi _{r'}\) of mass \(m_r\) can be added to it [6].
 4.
If the regularization preserves the vectorflavour symmetry and its conserved current is adopted, for instance, the corresponding variance vanishes in the infinite volume limit.
 5.
 6.
By the same argument, if a splitline estimator localized in a given region of space is chosen, the sum over \(\mathbf{y_2}\) in Eq. (5.10) is restricted to that region.
 7.
The so called oneend trick estimator used in the context of twistedmass discretization of QCD is a particular case of spliteven estimator, for which significant numerical gain has been observed empirically [17, 18]. The analysis of the variances presented here applies straightforwardly to this estimator too, for which a Schwarz inequality between its variance and the one of the standard estimator can also be derived.
 8.
Computing variances for such an optimization is cheap because it requires a few sources only.
 9.
We do not include the preparatory cost for computing \({\bar{t}}^{M}_{_{\Gamma ,r_n}}\) since it becomes quickly negligible after few evaluations of the randomnoise components of the estimator.
 10.
If we had used U(1) sources instead of Gaussian ones, the standard estimator for the pseudoscalar density would have a variance smaller by approximately a factor 3 on this lattice. We prefer to use Gaussian sources for all bilinears, however, for which the theoretical analysis is simpler.
 11.
 12.
In Ref. [31] probing vectors were introduced in the context of lattice QCD to define stochastic estimators of traces of the full quark propagator.
Notes
Acknowledgements
Simulations have been performed on the PC clusters Marconi at CINECA (CINECAINFN and CINECABicocca agreements) and Wilson at MilanoBicocca. We thank these institutions for the computer resources and the technical support. We are grateful to our colleagues within the CLS initiative for sharing the ensembles of gauge configurations with two dynamical flavours. L.G. and T. H. acknowledge partial support by the INFN project “High performance data network”.
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