through the matching [37]
˜
ϕ
M
ðx; μ
R
;P
z
Þ¼
Z
dyZ
ϕ
ðx; y; μ; μ
R
;P
z
Þϕ
M
ðy; μÞ
þ O
Λ
QCD
P
2
z
;
m
2
M
P
2
z
ð9Þ
according to LaMET. The quasi-DA can be obtained by
computing the following correlators for K
−
and η
s
,as
presented in the Refs. [16,17]:
C
2pt
ðz; P; tÞ¼h0j
Z
d
3
ye
i
P·y
¯
ψ
1
ð
y; tÞγ
z
γ
5
Y
z−1
x¼0
U
z
ðy þ xˆz; tÞ
× ψ
2
ð
y þ zˆz; tÞ
¯
ψ
2
ð0; 0Þγ
5
ψ
1
ð0; 0Þj0ið10Þ
where fψ
1
;ψ
2
gare fu;sgfor K
−
and fs;sgfor η
s
, Uð
x;
xþzÞ
is the W ilson line connecting lattice site
x to
x þ zˆz.
We perform a calculation using gauge ensembles with
clover valence fermions on a 48
3
×144 lattice with 2þ1þ1
flavors (degenerate up and down, strange, and charm
degrees of freedom) of highly improved staggered quarks
(HISQ) [38] generated by the MILC Collaboration [39].
The lattice spacing a ≈ 0.06 fm, and m
sea
π
¼ 310 MeV.
Hypercubic (HYP) smearing [40] is applied to the configu-
rations. The bare quark masses and clover parameters are
tuned to recover the lowest pion mass of the staggered quarks
in the sea. Correlators are calculated from momentum-
smearing sources [41] using 20 source locations on each of
the 95 configurations (1900 measurements in total).
We make two predictions using the ML algorithm. One
is to predict the correlators at larger link length z
pred
from
the correlators at z
in
<z
pred
. The other is to predict the
correlators of larger momentum p
pred
from the correlators
of p
in
<p
pred
.
To determine what input data to use for these predictions,
we first check the correlations among datasets with differ-
ent momenta, link lengths and time slices. The results are
shown in Fig. 1. Here, we set the target data to be the
2-point quasi-DA correlators at p
pred
¼ 5, z
pred
¼ 4 with
input data p
in
¼ 4, z
in
¼ 4 for p prediction and p
in
¼ 5,
z
in
< 4 for z prediction. We select the time slice t
pred
¼ 7 to
check the correlations.
Despite the larger error, larger time slices have a weaker
correlation with the target data. This suggests that we
should use input data close to the time slice of the target
data. On the other hand, we should be able to extend the
range of momentum or links of the input.
In the training process, we tried different parameters for
learning rate in f0.5; 0.2; 0.1; 0.02; 0.01; 0.005; 0.002g and
the number of estimators in f100; 150; 200; 250; 300g. The
corresponding fit variance are plotted in a heat map with
range [0,1], as shown in Fig. 2. Considering the fit quality
for both p predictions and z predictions, we selected
parameters r ¼ 0.1, N
est
¼ 150 as having highest fit quality
in both cases; these will be used for further meson-DA
predictions.
The datasets were evenly distributed into three parts:
training data, bias-correction data, and unlabeled test data.
In practice, we want to minimize the labeled data size
without sacrificing much prediction quality. We varied the
amount of training data and bias-correction data from 300
to 500, while keeping the number of unlabeled test data
N
ul
¼ 900 fixed, to look for a best trade-off between
reduced data size and prediction quality. The results are
shown in Fig. 3. When correlation is obvious, small number
of training and bias-correction datasets provides precise
estimate that is very close to the true observations for the
unlabeled dataset. When correlation is vague, the prediction
becomes more precise as one increases the size of the
training or the bias-correction datasets. Based on the plot,
we picked N
tr
¼ 400, N
BC
¼ 500 for further estimations.
FIG. 1. Correlations between target η
s
DA C
2pt
data at z
pred
¼ 4, p
pred
¼ 5, t
pred
¼ 7 with input data at a different link length
(momentum) and time slice for z prediction (left) and p prediction (right). The correlation decays quickly, especially at larger t.
ZHANG, FAN, LI, LIN, and YOON PHYS. REV. D 101, 034516 (2020)
034516-4