Precise determination of the lattice spacing in full lattice QCD

Abstract

We compare three different methods to determine the lattice spacing in lattice QCD and give results from calculations on the MILC ensembles of configurations that include the effect of $u$, $d$ and $s$ sea quarks. It is useful, for ensemble to ensemble comparison, to express the results as giving a physical value for $r_1$, a parameter from the heavy quark potential. Combining the three methods gives a value for $r_1$ in the continuum limit of 0.3133(23)(3) fm. Using the MILC values for $r_0/r_1$, this corresponds to a value for the $r_0$ parameter of 0.4661(38) fm. We also discuss how to use the $\eta_s$ for determining the lattice spacing and tuning the $s$-quark mass accurately, by giving values for $m_{\eta_s}$ (0.6858(40) GeV) and $f_{\eta_s}$ (0.1815(10) GeV). Comment: 15 pages

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Precise determination of the lattice spacing in full lattice QCD
C. T. H. Davies,1, ∗E. Follana,2I. D. Kendall,1G. Peter Lepage,3and C. McNeile1
(HPQCD collaboration), †
1Department of Physics and Astronomy, University of Glasgow, Glasgow, G12 8QQ, UK
2Department of Theoretical Physics, University of Zaragoza, E-50009 Zaragoza, Spain
3Laboratory of Elementary-Particle Physics, Cornell University, Ithaca, New York 14853, USA
(Dated: October 7, 2009)
We compare three different methods to determine the lattice spacing in lattice QCD and give
results from calculations on the MILC ensembles of configurations that include the effect of u, d
and s sea quarks. It is useful, for ensemble to ensemble comparison, to express the results as giving
a physical value for r1, a parameter from the heavy quark potential. Combining the three methods
gives a value for r1 in the continuum limit of 0.3133(23)(3)fm. Using the MILC values for r0/r1,
this corresponds to a value for the r0 parameter of 0.4661(38)fm. We also discuss how to use the
ηs for determining the lattice spacing and tuning the s-quark mass accurately, by giving values for
mηs(0.6858(40) GeV) and fηs(0.1815(10) GeV).
I. INTRODUCTION
Results from lattice QCD calculations are generally
computed in units of the lattice spacing a used in the
simulation. The lattice spacing must be computed sep-
arately and divided out in order to convert these results
into physical units (GeV, fm...), for comparison with
experiment. Any error in the lattice spacing determina-
tion feeds into most other quantities from lattice QCD,
and, in many cases, it is among the dominant sources
of errors. For example, in our determination of the de-
cay constant of the Dsmeson [1], 1% of the total error of
1.3% comes from the 1.5% uncertainty in the value of the
lattice spacing. Reducing the error on the lattice spacing
is then very important for increasing the precision of the
realistic lattice QCD calculations now possible [2].
Generally the value of the lattice spacing is determined
by comparing values from the simulation, in lattice units,
with values from experiment, in physical units. A lattice
simulation, for example, might give a value for the pion
decay constant in lattice units: aflat
perimental value fexp
π
in GeV gives a value for the lattice
spacing, a = (aflat
π
, in inverse GeV. This lattice
spacing can then be used to convert other simulation re-
sults from lattice units to physical units.
Lattice spacings determined in this way are inherently
ambiguous because lattice simulations are never exact.
In particular the use of a nonzero lattice spacing causes
lattice quantites, like flat
values, in this case fexp
π
. Such errors differ from quantity
to quantity, and therefore so will values for the lattice
spacing that are computed from these quantities. Such
differences, however, vanish in the continuum limit, a →
0, and so do not affect lattice predictions that have been
extrapolated to a = 0.
π. Dividing by the ex-
π)/fexp
π, to deviate from their physical
∗c.davies@physics.gla.ac.uk
†URL: http://www.physics.gla.ac.uk/HPQCD
In principle, any dimensionful quantity can be used to
determine the lattice spacing, but some quantities are
more useful than others. Ideally one wants quantities
that are easily computed, free of other types of simu-
lation error, largely independent of lattice parameters
other than the lattice spacing, and well measured in ex-
periments. Use of the pion decay constant, for example,
is not ideal. This decay constant is quite sensitive to the
u and d quark masses, which are generally too large in
current simulations; accurate values for the decay con-
stant can be obtained only after chiral extrapolations of
the simulation data to the physical quark masses. This
greatly complicates the use of the decay constant to set
the lattice spacing.
One physical quantity that is very easy to calculate
in lattice simulations is the r1 parameter derived from
the potential V (r) between two infinite-mass quarks sep-
arated by distance r. Parameter r1is defined implicitly
by the equation r2
C = 1 [3]. (Taking C = 1.65 gives the original such
standard parameter, r0[4].) This quantity is easily cal-
culated, in lattice units (that is, r1/a), to better than 1%.
Unlike the pion decay constant, it is only weakly depen-
dent upon the quark masses. It would be an ideal choice
for setting the lattice spacing except for the fact that
there is no experimental value for the physical r1—this
must be estimated instead from other lattice calculations.
In this paper we examine three other quantities that
can be used to determine the lattice spacing: 1) the
radial excitation energy in the Υ system (mΥ? − mΥ);
2) the mass difference between the Ds meson and one
half the ηcmass; and 3) the decay constant of the ficti-
tious ηsparticle, which can be related accurately to fK
and fπ. The valence-quark masses are easily tuned in
each case and each quantity is relatively insensitive to
sea-quark masses. Consequently each of these quantities
can be used to generate lattice spacings on an ensemble-
by-ensemble basis.
None of these quantities can be computed as accurately
as r1/a in simulations, but we can combine simulation re-
1F(r1) = C where F(r) ≡ dV/dr and
arXiv:0910.1229v1 [hep-lat] 7 Oct 2009

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sults for them with values for r1/a to obtain very accurate
estimates for the physical value of r1. Given r1, the differ-
ent values of r1/a can be used to obtain accurate lattice
spacings for each of the simulations we discuss here and
any other simulations where r1/a has been computed.
Of our three quantities, the ηs decay constant gives
the most accurate results. The ηsis a fictitious meson,
however, and so its “experimental” properties must be
related to those of real mesons using simulations. The
ηsis particularly closely related to the π and K mesons.
As we will show, its mass and decay constant can be
accurately related to those of the π and K through a
chiral analysis of simulation data for a variety of quark
masses and lattice spacings. Such an analysis also gives
an independent, fourth estimate of r1.
We describe in section 2 the three primary methods we
have used to obtain lattice spacings for a wide variety of
simulations. Each can be used to generate an estimate
for the physical value of r1, given values of r1/a.
section 3 we combine the three analyses to generate a
single, combined estimate for r1. This can then be used
to covert the r1/a values into a determination of a on
each ensemble. We also demonstrate how to determine
the lattice spacing from the ηswithout using r1. The two
methods are compared and shown to agree in the a → 0
limit. In Section 4 we give a value for r0derived from our
value of r1for comparison to others using that parameter.
In section 5 summarize our results. Finally, we discuss
the chiral analysis of decay constants and masses for the
π, K and their relation with those of the ηs meson in
Appendices A, B and C.
In
II. LATTICE CALCULATION
In Table I we list the parameter sets for the different
MILC ensembles of gluon configurations that we have
used here, although not all ensembles were used in every
lattice spacing determination.
Values for the static-quark potential parameter r1/a,
in lattice units, were determined by the MILC collabo-
ration [5]. They calculated the heavy quark potential by
fitting Wilson loops of fixed spatial size as a function of
lattice time. On the finest two sets of ensembles smeared
time links were used to reduce statistical noise and a two-
state exponential fit in time reduced the contamination
from excited potentials. The heavy quark potential ob-
tained was then fit as a function of spatial separation over
the range between 0.2fm and 0.7fm to a Cornell poten-
tial with the addition of corrections for lattice artifacts.
The point at which the condition for r1 held was then
determined from this fit. The errors given are statistical
errors only, since discretisation effects are taken care of
in our continuum extrapolations.
In what follows we will combine these values for r1/a
with estimates of the lattice spacing a determined using
three different physical quantities to obtain estimates for
the physical value of r1(that is, at zero lattice spacing
and with correct sea-quark masses).
A.mΥ? − mΥ
The calculation of the spectrum of mesons formed as
bound states of bottom quarks and antiquarks has been
an important test for lattice QCD. There are many radial
and orbital excitations below threshold for strong decay
and so many gold-plated states, well-characterised ex-
perimentally. The radial and orbital excitation energies
are almost identical for charmonium and bottomonium
when spin-averaged [6] and so rather insensitive to the
heavy quark mass. Heavy-quark vacuum polarization ef-
fects are tiny and so can be safely neglected. This makes
these systems very suitable for the determination of the
lattice spacing [7] and was one of the key calculations
demonstrating the importance of including the effect of
u, d and s sea quarks [2].
Here we improve on the calculations in [7] which used
results from MILC super-coarse, coarse and fine en-
sembles and compared ensembles with and without sea
quarks. We study only ensembles including sea quarks
but include also very coarse and superfine ensembles for
a wider range of lattice spacing values.
We calculate b-quark propagators on the MILC gluon
field configurations using lattice NonRelativistic QCD
(NRQCD) which has been developed over many years
to handle well the physics of heavy quark systems on the
lattice [8]. It makes a virtue of the nonrelativistic nature
of bottomonium bound states (v2
discarding the rest mass energy in favour of accurately
handling typical momentum and energy scales inside the
bound states. NRQCD can be matched to full QCD or-
der by order in v2
the nonrelativistic expansion and apply discretisation im-
provements through O(a2) to v2
magnetic and chromoelectric field-dependent terms at v4
(so that terms which induce fine structure are completely
improved to O(a4)). An analysis of remaining systematic
errors is given in [7].
The NonRelativistic Hamiltonian that we use is given
by [9]:
b≈ 0.1 for the Υ) by
band αs. We work through O(v4
b) in
bterms and to chromo-
b
aH = aH0+ aδH;
aH0 = −∆(2)
2aMb,
(∆(2))2
8(aMb)3+ c2
g
8(aMb)2σ ·
g
2aMb
a(∆(2))2
16n(aMb)2.
aδH = −c1
ig
8(aMb)2
?˜∇ טE −˜E ט∇
σ ·˜B + c5a2∆(4)
24aMb
?
∇ ·˜E −˜E · ∇
?
?
−c3
−c4
−c6
(1)
This is implemented in calculating b quark propagators

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3
TABLE I:
gauge coupling β, size L3× T and sea mass parameters masq
and masq
s
used for this analysis. The sea ASQTAD quark
masses (l = u/d) are given in the MILC convention where
u0 is the plaquette tadpole parameter. The lattice spacing
values in units of r1 after ‘smoothing’ are given in the third
column [5]. Sets 1 and 2 are ‘very coarse’; sets 3, 4 and 5,
‘coarse’; sets 6 and 7 ‘fine’; set 8 ‘superfine’ and set 9 ‘ultra-
fine’.
au0masq
1 6.5722.152(5)0.0097
2 6.586 2.138(4)0.0194
3 6.76 2.647(3)0.005
4 6.76 2.618(3)0.01
5 6.79 2.644(3) 0.02
6 7.09 3.699(3)0.0062
7 7.11 3.712(4)0.0124
8 7.46 5.296(7)0.0036
9 7.817.115(20) 0.0028
Ensembles (sets) of MILC configurations with
l
Setβr1/a
l
au0masq
0.0484
0.0484
0.05
0.05
0.05
0.031
0.031
0.018
0.014
s
L/a
16
16
24
20
20
28
28
48
64
T/a
48
48
64
64
64
96
96
144
192
by evolving the b quark Green’s function on a single pass
through the lattice using:
G(? x,t + 1) = (1 −aδH
2
)(1 −aH0
2n)n(1 −aδH
2n)nU†
t(x)
(1 −aH0
2
)G(? x,t) (2)
with starting condition:
G(? x,0) = φ(x)1.
(3)
Here ∇ is the symmetric lattice derivative and˜∇ is the
improved derivative,˜∇k = ∇k− ∇(3)
standard lattice discretisation of the second derivative
?
φ(x) is a real spatial smearing function which multi-
plies a unit matrix in color and spin space as the starting
point for the quark propagator. The antiquark propa-
gator for a given source is then the complex conjugate
of the Green’s function obtained from eq. 2. When the
quark and antiquark propagators are combined (with ap-
propriate Pauli matrices for different JPC[10]) into me-
son correlators, φ improves the overlap with particular
ground and excited states for a better signal. This will
be discussed further below.
n is a stability parameter which is chosen to tame (un-
physical) high momentum modes of the b quark propaga-
tor which might otherwise cause the meson correlators to
grow exponentially with time rather than fall. We used
n = 4 throughout this calculation instead of the value
n = 2 used earlier [7], so that we could work on finer
lattices and keep the same value of n for all ensembles.
This means that discretisation errors are smoothly con-
nected from one lattice spacing to another and the higher
value of n has the advantage of reducing some systematic
errors.
k/6.∆(2)is the
j∇(2)
mass in lattice units.
j
and ∆(4)is?
j∇(4)
j. aMbis the bare b quark
TABLE II: Parameters used in our calculations of b quark
propagators and bb correlators on various MILC ensembles,
numbered as in Table I. Mba is the bare b quark mass in
lattice units, u0L is the tadpole-improvement factor and the
stability factor n is taken as 4 everywhere. ncfg gives the
number of gluon field configurations used from the ensemble
and nt is the number of time sources for b quark propagators
per configuration. T is the time length in lattice units of the
propagators. a0 is the size parameter for the quark smearing
function φes(x) given in eq. 4.
Set
1
2
3
4
6
8
aMb
3.4
3.4
2.8
2.8
1.95
1.34
u0L
0.8218
0.8225
0.8362
0.8359
0.8541
0.8696
ncfg
631
631
2083
595
557
698
nt
24
24
32
32
8
8
T
32
32
32
32
48
48
a0
0.83
0.83
1.0
1.0
1.41
2.0
We use ‘tadpole-improvement’ [11] for all terms by di-
viding the gluon fields Uµby a factor u0when they are
read in. u0is taken as u0L, the value of the mean trace of
the gluon field for that ensemble in lattice Landau gauge
(where the trace is maximised). This removes, in a mean-
field way, the disparity between lattice and continuum
gluon fields induced by the fact that the lattice field is
exponentially related to the continuum field. Single com-
posite operators, such as ∆(4), are expanded out fully
so that all cancellations between U and U†are correctly
tadpole-improved. This is not done for the (1−aH0/2n)n
terms. Table II gives our parameters for the ensembles
used in the Υ analysis.
In order to reduce statistical errors over our previous
calculation we have investigated a number of improve-
ments. The first was to look at different forms for the
quark smearing φ(x). The simplest is a δ function but in
addition we can take an arbitrary functional form for φ(x)
provided that the gluon field configurations are gauge-
fixed, at least on a time-slice. The MILC configurations
that we use here are fixed to Coulomb gauge. When a
b quark propagator from a δ source and a b propagator
from a φ = f(x) source are combined a good overlap with
a particular Υ state is expected when, in the language of
a potential model, f(x) is a good approximation to the
wavefunction of that state. The ground state Υ(1S) will
dominate all 1−−correlators eventually so that there is
no advantage in including a smearing function that gives
a good overlap with that state [12]. Instead, to obtain a
good signal for the 2S−1S splitting, we concentrated on
functions that had very small overlap with the 1S state,
and therefore had better information about radial exci-
tations. A very good smearing for this was the function
from [7] called φes:
φes(r) = (2a0− r)exp(−r/(2a0)).
The size parameter, a0, was tuned on coarse lattices to
reduce the overlap of the correlator (known as the ‘ee’
correlator, see below) with the ground state, as judged
(4)

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4
by the small amplitude of the correlator at large times
when the ground state dominates. a0 was then scaled
as appropriate to ensembles of different lattice spacing.
Values are given in Table II. By combining b and b prop-
agators from δ function sources and φessources we are
able to make up 3 different meson smearing functions:
l is from combining two δ sources; e is from combining
a φes source with a δ source and E is from combining
two φessources (so that the composite meson smearing
function is then the convolution of φeswith itself). l, e
and E smearing functions can also be applied at the sink
to make a 3 × 3 matrix of correlators, with notation ll,
le, ee etc.
We also used a random wall source for our b quark
propagators, taking a set of U(1) random numbers, r,
with unit norm at every point on a time slice, one set
for each color of the b quark propagator. These were
combined with the smearing functions φ so that
?
When quark and antiquark propagators are combined to-
gether the random noise cancels except where the initial
spatial sites are the same and this effectively increases the
number of meson correlators sampled. We find that the
error on the ground state Υ energy is reduced by a factor
of 3 on coarse lattices and 5 on fine lattices, when cor-
rected to the same number of configurations. The excited
state energy does not improve by the same factor, how-
ever. Indeed we found rather little improvement in the
error on excited state energies which mirrors our experi-
ence with applying random wall sources to B mesons [13].
The inference is that random wall sources are much less
effective in situations where the degradation of the sig-
nal/noise is exponential. We calculate propagators from
many different time sources (which we then average over)
per configuration to improve statistical precision further.
The details of numbers of configurations and time sources
are collected in Table II.
As in [7] we use a Bayesian fitting method [14] to fit the
3×3 matrix of hadron correlators to a multi-exponential
form to extract the energies of states appearing in that
correlator. This alows us to fit the entire correlator (i.e.
for all time separations between source and sink), so mak-
ing use of all the information contained in it. It also
means that the fit results we obtain, for example for
the ground state, include the effect of the higher excited
states that are present in the correlator, and are not bi-
assed by an attempt to fit only one or two states in a
particular time window. The fitting function is
G(? x,0)c1c1=
? y
φ(|? x − ? y|)r(c1,? y)1spin.
(5)
Gmeson(nsc,nsk;t) =
nexp
?
k=1
a(nsc,k)a∗(nsk,k)e−Ekt. (6)
where a(nsc/sk,k) are the (real) amplitudes for state k
to appear in the smearings used at the source and sink
of the correlator respectively.
The Bayesian fitting method [14] allows a large num-
ber of exponentials to be used in the fit by constraining
the way in which these exponentials can appear based on
physical information. The simplest physical information
is that the energies of states are ordered, and we imple-
ment this in the fit by taking the energy fit parameters
as the natural logarithms of the ground state energy and
of the energy splittings between adjacent states. On top
of this we apply priors to the splittings between adjacent
states that constrain them to be of order 500MeV with
a width of a factor of two, i.e. between 250MeV and
1000MeV. Amplitudes are typically constrained around
zero with a width of 1.0 (our composite meson smearing
functions are normalised so that the spatial sum of their
square is 1). We apply a cut on the range of eigenvalues
present in the correlation matrix of 10−3except for the
high statistics calculation on the coarse 005/05 lattices
where we use 10−4. This reduces the number of degrees
of freedom in the fit to between 120 and 170, with 208 in
the coarse 005/05 fit. We obtain values for the Υ ground
state energy and that of the first radial excitation, the Υ?
as a function of the number of exponentials in the fits.
We demand a good χ2and that the fit for 3 adjacent
exponentials should agree both on the fitted values for
the energies of interest and on the errors. The ground
state energy stabilises very quickly, but the first excited
state is not generally stable until we reach 8 exponen-
tials. Fit results on the different MILC ensembles are
then tabulated in Table III from 10 exponential fits.
Figure 1 shows results from our highest statistics calcu-
lation on the coarse 005/05 lattices. Here we are able to
obtain a good signal for even higher excited states than
the 2S. The plot shows the ratio of the 3S −1S splitting
and the 4S − 1S splitting to that of the 2S − 1S. The
3S − 1S splitting is obtained to 3% and in agreement
with experiment. The 4S −1S splitting is not very accu-
rate even with the statistics available here. The result is
slightly higher than experiment, but the 4S state is not
gold-plated, decaying to BB. This is not taken account
of accurately in the lattice calculation and so we expect
our result to be higher than experiment. In our lower
statistics calculations we do not generally have a signifi-
cant signal for the 4S and our 3S − 1S splitting has an
error of between 5% and 10%.
As discussed earlier, the excitation energies for bound
states of heavy quarks are almost independent of the
heavy quark mass, meaning that accurate tuning of this
mass is not required for these splittings. Use of the ran-
dom wall does, however, allow us to determine the meson
energy as a function of meson momentum much more ac-
curately than in previous calculations, and so the meson
‘kinetic’ masses can be well determined. The meson mass
in NRQCD must be determined from the meson disper-
sion relation because the zero of energy has an offset.
The mass is then given by the difference in energy be-
tween mesons at zero momentum and momentum pa on

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5
1
1.5
2
2.5
3
4 6 8 10 12 14
nexp
(3S-1S)/(2S-1S)
(4S-1S)/(2S-1S)
expt
!2/dof
10 1.5 1.0 1.0 1.0 1.0 1.0
FIG. 1: Results for highly excited states from our fit to the
3 × 3 matrix of Υ correlators from the coarse 005/05 (set 4)
ensemble as a function of the number of exponentials included
in the fit. The χ2/dof is also shown - the fit had 208 degrees
of freedom. The results are stable from 9 to 12 exponentials.
the lattice by [7]:
Ma =p2a2− (∆Ea)2
2∆Ea
.
(7)
∆Ea is calculated by taking the difference in energy of
the ground state from a simultaneous fit to ll Υ corre-
lators made from a standard (zero momentum) random
wall as described above, and a random wall patterned
with an appropriate Fourier factor to give a momentum
of (1,0,0) to both quark and antiquark, so that the Υ
has momentum (2,0,0) (and its permutations). Values
obtained for the kinetic mass on each of the MILC en-
sembles are given in Table III. They are tuned within
10% of the experimental result of 9.46GeV [15].
Table III gives results for the lattice spacings aΥob-
tained by dividing the simulation results for a(E2− E1)
by the experimental value of 0.5630GeV for the split-
ting. Statistical errors are at the level of 1%. System-
atic errors arise from two sources, discretisation errors
and missing higher-order relativistic correctiosn to the
NRQCD action. The former can be removed by con-
tinuum extrapolation as long as they are well-behaved.
The leading discretisation corrections come from radia-
tive corrections to existing terms in the action and can be
calculated in perturbation theory. They have been shown
to be small corrections in the region of aMbin which we
work, and relatively independent of aMb[16]. Relativis-
tic corrections survive the continuum limit and are the
main source of systematic error for this method. They
TABLE III: Results for the ground state energy, aE1, and
radial excitation energy, a(E2− E1) obtained from 10 expo-
nential fits of the form in equation 6 to a 3 × 3 matrix of Υ
correlators as described in the text. The 4th column gives
the Υ mass as determined from eq. 7. Fewer configurations
were used for this than for the full calculation (and given in
Table II) in several cases. For set 3, 202 configurations were
used and for set 8, 470 configurations. The 5th column gives
the result for the lattice spacing from setting the 2S − 1S
splitting equal to the experimental value of 0.5630GeV [15].
Set
1
2
3
4
6
8
aE1
0.28775(8)
0.28814(8)
0.29330(3)
0.29261(6)
0.26618(5)
0.24850(3)
a(E2− E1)
0.4244(33)
0.4309(32)
0.3439(8)
0.3462(38)
0.2381(37)
0.1679(14)
aM
7.226(12)
7.231(12)
5.983(10)
5.985(11)
4.281(12)
3.050(18)
aΥ/fm
0.1488(12)
0.1510(11)
0.1205(3)
0.1213(13)
0.0835(13)
0.0588(5)
were estimated in [7] at 0.7% on the coarse and 0.6% on
the fine lattices, so we include an overall systematic error
of 0.7% in our error analysis here.
One ingredient missing from our calculation and
present in the experimental world is electromagnetism.
This is then another possible source of systematic error.
From a potential model calculation we estimate the shift
in the 2S − 1S splitting to be less than 1MeV from the
Coulomb interaction between b and b (the electromag-
netic self-interaction is included in the b quark mass).
At less than 0.2%, this is negligible.
To extract a physical value for the static-quark poten-
tial parameter r1, we must combine the lattice spacings
aΥ
Table I, and extrapolate to zero lattice spacing, correct-
ing the sea-quark masses. We do this by fitting (r1/a)iaΥ
from the ith ensemble to a formula for the effective r1
corresponding to mΥ? − mΥ:
rΥ
l
,δmsea
?

iin Table III with the corresponding values of (r1/a)iin
i
1(a,δmsea
s) = r1
(8)
×
1 + cΥ
sea
2δmsea
l
+ δmsea
ms
s
?
×
1 +
4
?
j=1
cΥ
j(a/r1)2j

,
jare the pa- where r1(the extrapolated value), cΥ
rameters tuned by the fit. Here the δmseaare differences
between the sea-quark masses used in the simulation and
the correct masses for l = u/d and s quarks (see Ap-
pendix C).
We have included twice as many terms as we need in
the expansion in a/r1; taking half as many terms gives
essentially identical results. We are able to retain higher-
order terms because we include Bayesian priors in our
fit for each expansion coefficient used—that is, we in-
clude an initial estimate for each parameter. Each prior
functions as an additional piece of input data in the fit,
thereby guaranteeing that we always have more fit data
seaand cΥ